Mike R. Jeffrey

Hidden

Dynamics

The Mathematics of Switches,

Decisions and Other Discontinuous

Behaviour

Hidden Dynamics

Mike R. Jeffrey

Hidden Dynamics

The Mathematics of Switches, Decisions

and Other Discontinuous Behaviour

123

Mike R. Jeffrey

Department of Engineering Mathematics

University of Bristol

Bristol, UK

ISBN 978-3-030-02106-1

ISBN 978-3-030-02107-8 (eBook)

https://doi.org/10.1007/978-3-030-02107-8

Library of Congress Control Number: 2018959419

Mathematics Subject Classification: 00-02, 03H05, 34E10, 34E13, 34E15, 34N05, 37N25, 37M99,

41A60, 92B99, 70G60, 34C23

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For Arthur.

A smooth sea never made a skillful sailor

– African proverb

Life has no smooth road for any of us;

and in the bracing atmosphere of a high aim

the very roughness stimulates the climber to steadier steps. . .

– William C. Doane

Artwork by Martin Williamson (http://www.cobbybrook.co.uk/) 2017

Preface

Discontinuities are encountered when objects collide, when decisions are

made, when switches are turned on or oﬀ, when light and sound refract as

they pass between diﬀerent media, when cells divide, or when neurons are

activated; examples are to be found throughout the modern applications of

dynamical systems theory.

Mathematicians and physicists have long known about the importance of

discontinuities. Studying light caustics (the intense peaks that create rainbows or the bright ripples in sunlit swimming pools), George Gabriel Stokes

lamented to his ﬁanc´ee in a letter from 1857:

. . . sitting up til 3 o’clock in the morning

. . . I almost made myself ill, I could not get over it

. . . the discontinuity of arbitrary constants.

Discontinuities are not a welcome feature in dynamical or diﬀerential equations, because they introduce indeterminacy, the possibility of one problem

having many possible solutions, many possible behaviours. How interesting

it is then to consider the thoughts of the inﬂuential engineer Ove Arup:

Engineering is not a science . . . its problems are under-deﬁned,

there are many solutions, good, bad, or indiﬀerent.

The art is . . . to arrive at a good solution.

For Arup, ‘science studies particular events to ﬁnd general laws’. Many mathematical scientists would agree that the goal is to achieve generality and

banish indeterminacy. But why should the two be mutually exclusive?

Unlocking the potential of discontinuities requires tackling these issues of

determinacy and generality. While accepting that some parts of the world lie

beyond precise expression, discontinuities nonetheless give us a way to express

them, to approximate that which cannot be approximated by standard ‘wellposed’ equations, and to explore a world where certain things will ever remain

hidden from view.

ix

x

Preface

With these ideas in mind, this book attempts to ready the ﬁeld of nonsmooth dynamics for turning to a wider range of applications, simultaneously

moving beyond the traditional scope of, and bringing our subject closer into

line with, the traditional theory of diﬀerentiable dynamical systems.

At a discontinuity, we lose access to some of the most powerful theorems

of dynamical systems, and it has long been the task of nonsmooth dynamical

theory to redress this. Progress has been impressive in some areas, limited

in others. We suggest here that much of what has gone before constitutes

a linear approach to discontinuities, and here, we lay the foundations for a

nonlinear theory. Making use of advances in nonlinearity and asymptotics,

once we can extend elementary methods such as linearization and stability

analysis to nonsmooth systems, discontinuities stop being objects of nuisance

and start becoming versatile tools to apply to modelling the real world.

Several examples of applications are studied towards the end of the book,

and many more could have been included. Interest in piecewise-smooth systems has been spreading across scientiﬁc and engineering disciplines because

they oﬀer reliable models of all manner of abrupt switching processes. Our

aim is to set out in this book the basic methods required to gain an in-depth

understanding of discontinuities in dynamics, in whatever form they arise.

In this book, a discontinuity is blown up into a switching layer, inside

which switching multipliers evolve inﬁnitely fast across the discontinuity. Several concepts may be at least partly familiar in other areas of mathematics,

in particular algebraic geometry, boundary layers, singularity theory, perturbation theory, and multiple timescales. The terminology used here does not

exactly correspond to the usage in those ﬁelds, and attempting to refer to or

resolve all of the clashes in nomenclature would not make for an easier read.

Moreover, we do not use the concepts themselves in strictly the same way.

For example, we use the idea of an inﬁnitesimal ε-width of a discontinuity

that we can manipulate algebraically, but we are interested solely in the limit

ε = 0. This proves to be a suﬃciently rich problem, and though it raises the

question of what happens when we perturb to ε > 0, that is left for future

work. As we discuss in Chapter 1 and Chapter 12, more so than in any smooth

system, the perturbation of a discontinuity is a many faceted problem.

This work builds on the pioneering eﬀorts particularly of Aleksei Fedorovich Filippov, Vadim I. Utkin, Marco Antonio Teixeira, and Thomas

I. Seidman. I have been lucky to meet and work with all but the ﬁrst of

these, and much in the spirit of modern science, they represent the truly international and interdisciplinary endeavour of what was for too long a niche

ﬁeld of study.

In essence, the framework introduced in this book seeks to explore Filippov’s world more explicitly, to make non-uniqueness itself a useful modelling

tool in dynamics. Sitting somewhere between deterministic dynamics and

stochastic dynamics, nonsmooth dynamics oﬀers a third way: systems that

are only piecewise-deﬁned, rendering them almost everywhere deterministic.

Bristol, UK

Mike R. Jeﬀrey

Chapter Outline

The book is roughly split into three parts: introductory material in Chapters 1

and 2, fundamental concepts at the level of the student or non-expert in

Chapters 3 to 6 and Chapter 14, and advanced topics in Chapters 7 to 13.

Chapter 1 is almost a stand-alone and informal essay, surveying the reasons why discontinuities occur, what forms they take, why they matter, and

how imperfect our knowledge of them is. The chapter is intended to provoke

thought and discussion, not to be detailed reference on the many theoretical

and applied concepts it touches on.

Chapter 2 is a stand-alone “lecture”-style outline, a crash course on the

topic, and a taster of the main concepts that will be developed in the book.

Chapter 3 contains the complete foundation for everything that follows,

the formalities for how we deﬁne piecewise-smooth systems in a solvable

way. This chapter contains the elements necessary for the eager researcher

to rediscover for themselves the contents of the remainder of the book and

beyond.

Chapter 4 sets out the basic themes that dominate piecewise-smooth dynamics, the kinds of orbits, the key singularities, and the concepts of stability

and bifurcation theory.

Chapter 5 deﬁnes a general prototype expression for piecewise-smooth

vector ﬁelds in the form of a series expansion.

Chapter 6 describes the basic forms of contact between a ﬂow and a discontinuity threshold.

Chapter 7 contains the most important new theoretical elements of the

book, setting out the analytical methods required to understand piecewisesmooth systems.

Chapter 8 takes a step back, applying the previous chapters in the more

standard setting of linear switching.

Chapter 9 begins the leap forward into nonlinear switching, revealing some

of the novel phenomena of piecewise-smooth systems.

xi

xii

Chapter Outline

Chapter 10 focusses on the most extreme consequences of discontinuity,

via determinacy breaking and loss of uniqueness.

Chapter 11 tackles how we understand large-scale behaviour, with new

notions of global dynamics and associated bifurcations.

Chapter 12 asks how robust everything that has come before is. We consider how our mathematical framework can be interpreted in a practical setting and what happens to it in the face of nonideal perturbations.

Chapter 13 visits an old friend and long-term obsession of piecewisesmooth systems, the two fold singularity.

Chapter 14 is a series of case studies applying the foregoing analysis to

‘real-world’ models.

Exercises are provided at the back of the book to further facilitate a more

in-depth reading or lecture course.

How to Use This Book

This book will look rather diﬀerent to other works in the area. In Chapter 1,

we start from a tour of some less quoted, wide-ranging, but fundamental,

examples of how discontinuity arises. Chapter 3 presents the formalism for

studying nonsmooth dynamics that forms the foundation for everything that

follows and should be the starting point for any course. It is quite possible

to jump from there to Chapter 12 to focus on the application and robustness

of the formalism.

A proper understanding of the dynamics of nonsmooth system, or a course

in it, should progress through Chapters 4 to 11, and I would suggest focussing

on (and indeed extending) the analytical methods in Chapter 7. The great

peculiarities of nonsmooth systems begin to be revealed in Chapter 9 and

Chapter 10, and there are numerous examples therein to explore and build on.

More in-depth applications are given in the form of case studies in Chapter 14.

Exercises provided at the back of the book provide further insight into the

various examples and theorems explored, chapter by chapter.

Prerequisites. In reading this book, it will be helpful to have a grounding in (though we give elementary introductions where possible): single and

multi variable calculus, Taylor series, ordinary diﬀerential equations and elementary dynamical systems, some linear algebra (eigenvectors, etc.), and a

little introductory (highschool) physics. Applications will be explained with

background where they are discussed.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

1

Origins of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Discontinuities and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Discontinuities and Determinism . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Discontinuities in Approximations . . . . . . . . . . . . . . . . . . . . . . .

1.4 Discontinuities in Physics and Other Disciplines . . . . . . . . . . .

1.4.1 In Mechanics: Collisions and Contact Forces . . . . . . . .

1.4.2 In Optics: Illuminating a Victorian Discontinuity . . . .

1.4.3 In Sound: Wavefronts and Shocks . . . . . . . . . . . . . . . . . .

1.4.4 In Graphs: Sigmoid Transition Functions . . . . . . . . . . .

1.5 ‘Can’t We Just Smooth It?’ (And Other Fudges) . . . . . . . . . .

1.6 Discontinuities and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7 Discontinuity in Dynamics: A Brief History . . . . . . . . . . . . . . .

1.8 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

5

7

10

12

14

15

18

20

22

26

30

2

One

2.1

2.2

2.3

2.4

31

31

37

38

41

47

48

48

50

52

53

56

2.5

2.6

2.7

Switch in the Plane: A Primer . . . . . . . . . . . . . . . . . . . . . . .

The Elements of Piecewise-Smooth Dynamics . . . . . . . . . . . . .

The Value of sign(0): An Experiment . . . . . . . . . . . . . . . . . . . . .

Types of Dynamics: Sliding and Crossing . . . . . . . . . . . . . . . . .

The Switching Layer and Hidden Dynamics . . . . . . . . . . . . . . .

2.4.1 A Note on Modelling Basic Oscillators . . . . . . . . . . . . .

Local Singularities and Bifurcations . . . . . . . . . . . . . . . . . . . . . .

2.5.1 Equilibria and Local Stability . . . . . . . . . . . . . . . . . . . . .

2.5.2 Tangencies and Their Bifurcations . . . . . . . . . . . . . . . . .

2.5.3 Equilibria, Sliding Equilibria, and Their Bifurcations .

Global Bifurcations and Tangencies . . . . . . . . . . . . . . . . . . . . . .

Determinacy-Breaking: A First Glimpse . . . . . . . . . . . . . . . . . .

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Contents

2.8

2.9

Counting Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3

The

3.1

3.2

3.3

3.4

3.5

3.6

Vector Field: Multipliers and Combinations . . . . . . . . . .

Piecewise-Smooth Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . .

The Discontinuity Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Constituent Fields and Indexing . . . . . . . . . . . . . . . . . . . . . . . . .

The Switching Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Inclusions and Existence of a Flow . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

61

62

63

67

69

72

4

The

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Flow: Types of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Types of Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Determinacy Breaking Events . . . . . . . . . . . . . . . . . . . . . . . . . . .

Equivalence and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Flow Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

73

74

79

83

88

89

90

5

The Vector Field Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 The Vector Field Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 The Canopy for One Switch . . . . . . . . . . . . . . . . . . . . . . .

5.1.2 The Canopy for Two Switches . . . . . . . . . . . . . . . . . . . . .

5.1.3 The Canopy for m Switches . . . . . . . . . . . . . . . . . . . . . . .

5.2 Deriving the Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Joint Expansions and Matching . . . . . . . . . . . . . . . . . . .

5.2.2 Series of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.3 Uniqueness of the Multilinear Term . . . . . . . . . . . . . . . .

5.3 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

91

92

93

94

95

95

96

98

100

6

Tangencies: The Shape of the Discontinuity Surface . . . . . .

6.1 Flow Tangencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Fold (d = 2, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Two-Fold (d = 3, k = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 Cusp (d = 3, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5 Swallowtail (d = 4, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 Umbilic: Lips and Beaks (d = 4, k = 2) . . . . . . . . . . . . . . . . . . .

6.7 Fold-Cusp (d = 4, k = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8 Many-fold Singularities, Cusp-Cusps, and So On . . . . . . . . . . .

6.9 Proofs of Leading-Order Expressions for the Fold and

Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10 A Note on Alternative Classiﬁcations . . . . . . . . . . . . . . . . . . . . .

6.11 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

103

109

110

112

114

115

117

118

119

123

124

Contents

xv

7

Layer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 The Sliding Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.1 A Special Case: ‘Higher-Order’ Sliding Modes . . . . . . .

7.2 The Sliding Region’s Attractivity . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Singularities of the Sliding Manifold M . . . . . . . . . . . . . . . . . . .

7.4 End Points of the Sliding Region . . . . . . . . . . . . . . . . . . . . . . . .

7.5 Multiplicity and Attractivity of Sliding Modes . . . . . . . . . . . . .

7.5.1 One Switch, Multiple Sliding Modes . . . . . . . . . . . . . . .

7.5.2 Multiplicity of Sliding Modes at Intersections . . . . . . .

7.5.3 Classiﬁcation of Sliding Modes/Equilibria . . . . . . . . . . .

7.6 Layer Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.7 Equilibria and Sliding Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .

7.8 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9 Bifurcations of Equilibria and Sliding Equilibria . . . . . . . . . . .

7.10 A Saddlenode/Persistence Criterion . . . . . . . . . . . . . . . . . . . . . .

7.11 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

126

131

131

134

138

140

141

143

146

149

154

159

160

168

168

8

Linear Switching (Local Theory) . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1 The Sliding Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2 The Convex Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3 Equilibria and Sliding Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .

8.4 Boundary Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . . . .

8.4.1 One-Parameter BEBs in the Plane . . . . . . . . . . . . . . . . .

8.5 Boundaries of Sliding: For a Single Switch . . . . . . . . . . . . . . . .

8.5.1 Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.2 Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.3 Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.4 Swallowtail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.5 Umbilic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.6 Fold-Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6 Bifurcations of Sliding Boundaries in the Plane . . . . . . . . . . . .

8.7 Boundaries of Sliding: For r Switches . . . . . . . . . . . . . . . . . . . . .

8.8 The Hidden Degeneracy of Linear Switching . . . . . . . . . . . . . .

8.9 Piecewise-Smooth Time Rescaling . . . . . . . . . . . . . . . . . . . . . . .

8.10 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

172

173

174

175

175

180

182

183

185

186

188

190

192

195

198

199

200

9

Nonlinear Switching (Local Theory): The Phenomena of

Hidden Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 Nonlinear Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 Hidden Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.1 A Hidden van der Pol Oscillator . . . . . . . . . . . . . . . . . . .

9.2.2 Hidden Duﬃng Oscillator and Ueda chaos . . . . . . . . . .

9.2.3 Cross-Talk Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.4 Hidden Lorenz Attractor . . . . . . . . . . . . . . . . . . . . . . . . .

201

201

204

204

206

208

211

xvi

Contents

9.3

Hidden Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1 Cross or Not at an Intersection . . . . . . . . . . . . . . . . . . . .

The Illusion of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.1 Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.2 Slip Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.3 Switching with Time Dependence . . . . . . . . . . . . . . . . . .

Nonlinear Switching as a Small Perturbation . . . . . . . . . . . . . .

9.5.1 Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.2 Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.3 Structural Stability of the Sliding Manifold . . . . . . . . .

Hidden Degeneracy at Local Bifurcations . . . . . . . . . . . . . . . . .

9.6.1 Boundary Node Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.2 Fold-Fold and Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213

213

215

216

217

221

223

225

226

227

231

231

236

241

10 Breaking Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.1 Exit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.2 Exit Points: Deterministic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.2.1 Exit via a Simple Tangency . . . . . . . . . . . . . . . . . . . . . . .

10.2.2 Exit Transverse to an Intersection . . . . . . . . . . . . . . . . .

10.2.3 Exit via Tangency to an Intersection . . . . . . . . . . . . . . .

10.3 Exit Points: Determinacy-Breaking . . . . . . . . . . . . . . . . . . . . . .

10.3.1 Exit Transverse to an Intersection . . . . . . . . . . . . . . . . .

10.3.2 Exit via a Complex Tangency . . . . . . . . . . . . . . . . . . . . .

10.3.3 Zeno Exit from an Intersection . . . . . . . . . . . . . . . . . . . .

10.3.4 Exit from a Sliding Fold . . . . . . . . . . . . . . . . . . . . . . . . . .

10.4 Stranger Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.5 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243

243

244

244

245

247

250

251

257

263

269

270

272

11 Global Bifurcations and Explosions . . . . . . . . . . . . . . . . . . . . . . .

11.1 Local Classiﬁcation of Global Phenomena . . . . . . . . . . . . . . . . .

11.2 The Sliding Eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.3 Sliding Bifurcations/Explosions: The Global Picture . . . . . . .

11.4 Sliding Bifurcations/Explosions in Nonlinear Switching . . . . .

11.5 The Classiﬁcation and Its Completeness . . . . . . . . . . . . . . . . . .

11.5.1 Unfoldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.5.2 Classes of Sliding Bifurcation . . . . . . . . . . . . . . . . . . . . .

11.5.3 Classes of Sliding Explosion . . . . . . . . . . . . . . . . . . . . . .

11.5.4 The Omitted Singularities . . . . . . . . . . . . . . . . . . . . . . . .

11.6 Codimension Two Sliding Bifurcations and Explosions . . . . . .

11.7 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.8 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273

273

276

279

286

289

291

292

298

302

302

303

305

9.4

9.5

9.6

9.7

Contents

xvii

12 Asymptotics of Switching: Smoothing and Other

Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1 Probabilistic Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1.1 Multiplying Probabilities in the Combination . . . . . . .

12.1.2 The Unreasonable Eﬀectiveness of Nonsmooth

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.2 Convex Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.2.1 Experiments on Convex Switching . . . . . . . . . . . . . . . . .

12.2.2 Conclusion: Jitter Over the Convex Hull . . . . . . . . . . . .

12.3 Smooth Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.1 Why Smooth? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.2 The Smoothing Tautology . . . . . . . . . . . . . . . . . . . . . . . .

12.3.3 Deriving the Layer System via Smoothing . . . . . . . . . .

12.3.4 Equivalence of the Smoothed System? . . . . . . . . . . . . . .

12.3.5 Equivalence of Layer Dynamics . . . . . . . . . . . . . . . . . . . .

12.3.6 The Degeneracy of L Persists to L . . . . . . . . . . . . . . . .

12.3.7 Exponential Sensitivity, Contraction, and

Determinacy-Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.8 The Canopy as a Series Expansion . . . . . . . . . . . . . . . . .

12.4 Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.1 Extrinsic Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.2 Intrinsic Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.5 Intermediary Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.6 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

336

337

340

343

346

350

352

13 Four Obsessions of the Two-Fold Singularity . . . . . . . . . . . . . .

13.1 The Generic Two-Fold: A Summary . . . . . . . . . . . . . . . . . . . . . .

13.2 Obsession 1: The Prototype in n Dimensions . . . . . . . . . . . . . .

13.2.1 The Visible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.2.2 The Visible-Invisible Two-Fold . . . . . . . . . . . . . . . . . . . .

13.2.3 The Invisible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.2.4 Geometry of the Angular Jump Parameter ν + ν − . . . .

13.3 Obsession 2: The Nonsmooth Diabolo . . . . . . . . . . . . . . . . . . . .

13.3.1 First Return Map: The Skewed Reﬂection . . . . . . . . . .

13.3.2 The Rotation Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.3 Number of Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.4 The Nonsmooth Diabolo . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.5 The Nonsmooth Diabolo Bifurcation . . . . . . . . . . . . . . .

13.4 Obsession 3: The Folded Bridge . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.1 The Visible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.2 The Visible-Invisible Two-Fold . . . . . . . . . . . . . . . . . . . .

13.4.3 The Invisible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.4 The Nonsmooth Diabolo Bifurcation: Sliding . . . . . . . .

355

356

359

361

362

363

365

367

367

369

371

378

380

385

387

388

389

390

307

307

309

311

314

314

320

321

321

322

328

329

329

335

xviii

Contents

13.5 Obsession 4: Sensitivity in the Layer . . . . . . . . . . . . . . . . . . . . .

13.5.1 The Unperturbed System . . . . . . . . . . . . . . . . . . . . . . . . .

13.5.2 The Perturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.6 An Unﬁnished Saga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

392

393

395

403

14 Applications from Physics, Biology, and Climate . . . . . . . . . .

14.1 In Control: Steering a Ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.2 Ocean Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3 Chaos in a Church . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.1 ‘Lumped Water’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.2 ‘Moving Point’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.3 ‘Discrete Kick’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.4 Explosion in a Superconducting Stripline Resonator . . . . . . . .

14.5 Conical Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.6 Optical Folded Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.7 Static Versus Kinetic Friction . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.8 A Paradox of Skipping Chalk . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.9 Pinching Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.9.1 Relaxation Oscillations and Canards . . . . . . . . . . . . . . .

14.9.2 Local Geometry of the Canard Singularity . . . . . . . . . .

14.9.3 The First Pinch: A Shot in the Dark . . . . . . . . . . . . . . .

14.9.4 The Second Pinch: Zooming in on the Manifolds . . . . .

14.9.5 The Third Pinch: Catching the Canards . . . . . . . . . . . .

14.10 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407

407

409

418

419

421

422

424

432

437

442

449

455

456

459

463

464

466

473

A

Discontinuity as an Asymptotic Phenomenon: Examples . .

A.1 Changes of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1.1 Large-Scale Bistability, Small-Scale Decay . . . . . . . . . .

A.1.2 Large-Scale Bistability, Small-Scale Dissipation . . . . . .

A.2 In Integrals: Stokes’ Phenomenon . . . . . . . . . . . . . . . . . . . . . . . .

A.3 In Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475

475

475

477

478

480

B

A Few Words from Filippov and Others, Moscow 1960 . . . 481

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Chapter 1

Origins of Discontinuity

Discontinuities occur when light refracts, when neurons or electronic switches

activate, and when collisions or decisions or mitosis or myriad other processes

enact a change of regime. We observe them in empirical laws, in the structure

of solid bodies, and also in the series expansions of certain mathematical

functions.

As commonplace as they may be, discontinuities are a curious thing to

try to build into dynamical models. They violate that central requirement of

calculus: to be continuous. They permit determinism to collapse in ﬂeeting

bursts of non-uniqueness. They conjure up a new realm of nonlinear dynamics.

In this ﬁrst chapter we start by exploring what it means for a system to be

discontinuous. Some discontinuities we understand, but many, the kinds of

discontinuities we encounter in engineering, the life sciences, and economics

and desperately wish to develop more sophisticated models for, we hardly

understand at all.

Formally, this book asks what happens to the trajectories of variables

x(t) = ( x1 (t), x2 (t), . . . , xn (t) ) ∈ Rn ,

(1.1)

as they evolve in time t, according to a set of ordinary diﬀerential equations

d

x = f (x) ,

dt

(1.2)

when the right-hand side is only piecewise-smooth, changing smoothly with

respect to x almost everywhere, except at certain thresholds σ(x) = 0 where

the value of f jumps, i.e. is discontinuous. But this rather dry statement hints

at few of the pitfalls and paradoxes of dynamics aﬄicted by discontinuities.

© Springer Nature Switzerland AG 2018

M. R. Jeﬀrey, Hidden Dynamics,

https://doi.org/10.1007/978-3-030-02107-8 1

1

2

1 Origins of Discontinuity

1.1 Discontinuities and Dynamics

When Isaac Newton set down the laws of motion that form the basis of

classical mechanics, he helpfully also set out the route to understand them

using calculus. Yet in doing so he mischievously threw into the stirring pot

some laws of motion not amenable to calculus. Century upon century since, a

juxtaposition of continuous and discontinuous change at the heart of physics

has remained, with consequences that remain only partly understood.

Collisions oﬀer a tangible example (Figu m

ure 1.1). Newton’s laws tell us the forces

v

n

acting on a moving object, and from those

lisio

M

-col

e

r

p

forces, calculus provides its speed and position. Yet when that object collides with another, instead of calculus we must employ a

M m

little mathematical sleight of hand. Calcun

isio

lus works for the pre-collision motion, and

coll

it works for the post-collision motion, but

then we must stitch the two together someu’

what artiﬁcially. To disguise the conceit—

m

n

v’ M

the discontinuity in the laws of motion—we

llisio

t-co

s

o

p

give the procedure a lofty title: an impact

law.

Fig. 1.1 Two objects collide and

Discontinuities allow us to gloss over

recoil. An impact law relates their

small details that seem to have no major

incoming speed u + v to their reeﬀect on our large-scale view. The last cencoil speed u + v by u + v =

e(u + v) for some 0 ≤ e ≤ 1.

tury, however, has taught us that no matter

how small, details can change everything.

The reason that we cannot follow motion through a collision, in the same

way we can follow objects that are rolling or in free ﬂight, is because the collision involves stepping between irreconcilable physical regimes: free motion

and rigid contact. One way to understand the regime change is to step into a

diﬀerent modelling approach entirely, perhaps on a ﬁner scale allowing bodies

to be more compliant and less idealized. But this can bring its own problems

and ambiguities, introducing much greater complexity, often probing areas

where our knowledge is less complete, and ultimately being diﬃcult to marry

up with the original discontinuous model.

To serve those situations, our task in this book, and in the ﬁeld of nonsmooth or piecewise-smooth dynamics more widely, is to provide a way within

a given dynamical model, to follow motion across the discontinuities between

irreconcilable regimes.

As science spreads its interest to new technological and sociological vistas,

it increasingly encounters a world full of irreconcilable regimes, of media not

behaving like steady waves rolling over the ocean, like electromagnetic waves

vibrating through spacetime, or like spheres orbiting and tumbling through

the vacuum of the heavens. Instead we ﬁnd abrupt changes that we patch

1.1 Discontinuities and Dynamics

3

reflectivity

over with ad hoc rules, such as switch

from behaviour A to behaviour B. Figure 1.2, for example, shows a discontinuity that turns up in climate models—

the reﬂectivity of the Earth’s surface

jumping across the edge of an ice shelf.

ce

The mathematical implications of such

tan

s

i

d

switches are not obvious.

Discontinuities like these are what

endow the world around us with struc- Fig. 1.2 A jump in surface reflectivity

between ice and water oceans.

ture. The boundaries of solid objects are

marked by jumps in properties like density, elasticity, or reﬂectivity. People

make decisions changing the course of their day. Storms and waves and glasses

break, social regimes change, lives are stopped and started.

As with collisions, we tend to skirt around the edges of these discontinuities

with a little sleight of hand and so describe almost everything going on in

a system, glancing over the discontinuities which, after all, are but ﬂeeting.

When I choose to go left or right, when a cell chooses to grow or divide, and

when a machine switches on or oﬀ—that brief moment when the choice in

enacted is trivial, isn’t it?

Far from it. Three centuries of calculus have left mathematicians uneasy

with discontinuities and reluctant to give up the continuity that provides

so many theorems concerning stability, attractors, bifurcations, and chaos,

because discontinuities leave these theorems in tatters.

From the mathematical point of view, a discontinuity renders a system

‘ill-posed’. A well-posed system has equations whose solutions: (i) exist, (ii)

are unique, and (iii) vary continuously with initial conditions. To satisfy all

three, a system must be smooth enough (meaning diﬀerentiable some number

of times, and certainly anything with a discontinuity does not qualify).

It turns out that at discontinuities we will often have to give up properties

(ii) and (iii), but not (i), not existence. It may seem perverse to give up

uniqueness and continuous dependence on initial conditions, but that is what

discontinuities are, events by which continuity and uniqueness are lost, and

our task is not to judge, but to learn how those losses can be exploited to

understand more about the world around us.

This book is an exploration of that idea. It is an attempt to extend

the methods of nonlinear dynamics beyond the barriers that discontinuities

have previously made impassable. In pushing back these boundaries, we ﬁnd

some intriguing behaviours. The methods, the theory behind them, and the

phenomena we discover, all require deeper future study. Though we prove

results where possible, not everything we do can be elevated to the level

of rigour that can be achieved with smooth systems (at least not yet), so

we do not claim a rigorous study here, only a development of ideas and

methods.

4

1 Origins of Discontinuity

Throughout the book we study the discontinuous system, with all of

the diﬃculties that brings, breaking only in Chapter 12 to consider nearby

‘perturbations’. There are various obvious ways that one may try to avoid

discontinuity depending on context. We might, for example, smooth out a

discontinuity, perhaps believing that smooth physical laws underlie it or

simply to make it easily computable. Or we might blur the discontinuity

with a distributive or stochastic process. An entire book mirroring this one

could be written using each approach, one smooth and deterministic and one

stochastic.

The discontinuous approach accepts that either of these, or numerous other

perturbations of the discontinuous model, could be the right approach. Let

us ﬁrst attempt to understand the underlying discontinuity, and later we will

probe a little into what happens when we perturb, in one way or another, by

smoothing, randomizing, or blurring the discontinuity in other ways.

The book starts and ends with less formal chapters which set the context

for our subject matter with the use of practical examples. This is one such

chapter and takes us on a short tour of how discontinuities arise and some

phenomena they produce. This expedition is not vital for those seeking an

introduction to piecewise-smooth dynamical systems theory, nor is it a comprehensive study of the topics touched on, but I hope you will at least skim

through it as motivation for what is to come.

In between those less formal chapters come more technical theory, aimed

at developing methods to understand the geometry and stability of solutions,

rather than focussing on proofs of solvability and universality of classes, but

opening numerous avenues for future study. After the theory is established

in Chapters 3 to 7 and explored at little in Chapters 8 to 11, we delve more

deeply into applications and ‘real-world’ switches in Chapters 12 to 14.

Towards the end of the book, we return to the question of what a discontinuity is. Discontinuities allow us to model abrupt change without imposing

undue structure. In a story that will unfurl as we reach Chapter 12, we will

learn that the best achievable representation of reality is not always the most

precise. We will see that it is sometimes unuseful, and even misleading, to

model processes in ﬁner detail than our understanding allows and that discontinuities provide not an obstacle to calculus but a new vehicle for it to

traverse uneven terrain.

To rely on continuity is to overlook that discontinuities are inescapable.

They arise not only in our everyday reality but within calculus itself, in the

midst of divergent series and singular perturbations, leaving mathematics no

less rich or rigorous for it. To rely on continuity is to risk overlooking that

diﬀerentiability reaches only so far into the complexities of a real world where

discordant media interact over disparate scales, and discontinuities are often

the result. We visit all of these in this chapter.

So let us see why discontinuity matters, where it comes from, and what it

looks like.

1.2 Discontinuities and Determinism

5

1.2 Discontinuities and Determinism

One issue will concern us only in limited situations, but will not go away

altogether, and that is:

where there are discontinuities there is non-uniqueness.

This non-uniqueness comes in many guises, but with just two main sources

that we can introduce brieﬂy.

The ﬁrst comes from a lack of knowledge of what happens inside a discontinuity. We may know that a quantity jumps between two values, but not

know precisely how it does so. We then use hidden terms to bring this uncertainty to life, to express the diﬀerent possible modes of behaviour inside the

jump. We shall show these constitute a form of nonlinearity. This is one of

the more subtle notions that will unfold throughout this book, and we will

introduce them a little more in Section 1.3.

The second source of non-uniqueness is more obvious, more well known,

and is the reason why mathematicians are taught a reluctance to study nonsmooth systems. It aﬄicts the solutions of a diﬀerential equation at a discontinuity. A classic example is the equation

dx

= |x|α ,

dt

(1.3)

for diﬀerent values of α ≥ 0. Its solutions take the form

x(t) = x0 1 +

1−α

t

x0 |x0 |−α

1/(1−α)

,

(1.4)

with an initial condition x(0) = x0 . Although we can write the solution

(fairly) simply, upon closer inspection we start to ﬁnd problems with it.

For α ≥ 1 solutions come in three types: those that start at x0 = 0 and sit

there forever, those that start at x0 < 0 and tend to x = 0 but never quite

reach it, and those that start at x0 > 0 and head oﬀ towards inﬁnity. For

instance, in the special case α = 1, we simply have dx

dt = |x|, and the solutions

become x(t) = x0 esign(x0 )t . The solution through any x0 is therefore unique: if

we know the ‘x0 ’ where we start, then all future (or indeed past) evolution of

x(t) is determined. This follows from the continuity of |x|α for α ≥ 1 (more

of

precisely the Lipschitz continuity of |x|α , by the so-called Picard-Lindel¨

theorem [149]).

For 0 < α < 1 the situation is entirely diﬀerent. The discontinuity in the

derivative of (1.3) takes over. Every solution through any x0 < 0 reaches

x = 0 in a future time t = |x0 |1−α /(1 − α), while every solution through any

x0 > 0 must have left x = 0 at a past time t = −|x0 |1−α /(1 − α). Does this

mean that we just have one solution that passes through zero? No, because

the point x = 0 is a solution itself. So if a solution from x < 0 reaches x = 0,

6

1 Origins of Discontinuity

it can sit there arbitrarily long before setting oﬀ again towards x > 0. This

means that an inﬁnity of diﬀerent solutions, all pausing to rest for diﬀerent

amounts of time at x = 0, all overlap at the origin and we cannot tell them

apart. As a result, the history and future of the point x0 = 0 are non-unique.

Non-unique histories are part of everyday experience and are one of the

reasons why nonsmooth systems have such broad applications. For example,

imagine an object that has been propelled along a surface and brought to rest

by friction. It is subsequently impossible to reconstruct the object’s motion

before it came to rest or to determine how much time has elapsed since it

stopped. A discontinuity in the frictional interaction between the object and

the surface has destroyed this information. This is an important eﬀect in our

everyday lives. When you hit the brakes in your car, you want them to behave

like 0 < α < 1 in the example above, to come to rest in ﬁnite time, not to

slow interminably towards the scene of an accident.

Non-unique futures are something less comfortable. A solution can start

out being unique and well behaved, but in the presence of a discontinuity, it

can ﬁnd itself ripped apart and endowed with inﬁnitely many possible futures.

We call these determinacy-breaking events.

Figure 1.3 depicts the scenario schematically. The picture shows the trajectories of a system evolving through space. Those trajectories are deterministic

everywhere except at a single point, the determinacy-breaking singularity.

Exit

trajectories

Inset

E I

determinacy

-breaking

Fig. 1.3 A determinacy-breaking event. Solutions before and after the singularity are

deterministic. Any trajectory starting in I hits the singularity. All trajectories in E originate

at the singularity. Inset right: forming a closed set.

Such singularities are common in nonsmooth systems. They result in new

kinds of nonlinear dynamics, new kinds of chaos and bifurcations, and even

new kinds of attractors. Imagine in Figure 1.3, for instance, if the inset I of

trajectories that are pulled into the singularity is intersected by the exit set

E of trajectories leaving the singularity (shown inset right). Then trajectories

will exist that make repeated yet unpredictable excursions, trapped forever

to return to the singularity, despite their exit path from it being uncertain.

With its inherent ambiguities of various sources of non-uniqueness, it is

easy to dismiss discontinuities from serious dynamical theory. But the nonuniqueness turns out to be useful, not to be swept under the rug or axiomatized into oblivion, and closely intwined in all its forms with nonlinearity.

1.3 Discontinuities in Approximations

7

1.3 Discontinuities in Approximations

How do you approximate near a discontinuity? This is what we are doing

very often when we are studying discontinuous systems and their dynamics, whether in theoretical equations or in empirical models. Consider the

following.

Example 1.1 (Approximating a Nonlinear Switch). Let us try to approximate

a pair of functions

g(x) =

sin x

|x|

and

2

f (x) = (1 + 2g(x)) ,

(1.5)

sketched in Figure 1.4. (In a strict sense we should not refer to f and g as

functions if they take many values at x = 0, but we allow this small abuse

of terminology, much as the Heaviside step ‘function’ or sign ‘function’ are

so-called, with the values at x = 0 being, after all, our topic of interest).

+1

9

g(x)

0

f(x)

−1

0

x

1

0

0

x

Fig. 1.4 The graph of two functions g(x) and f (x) with a discontinuity at x = 0.

These are both well behaved for x away from zero, and if we wish to approximate them near a point c = 0, we can expand them as Taylor series,

g(x) =

f (x) =

sin c

|c|

c−sin c

+ (x − c) c cosc|c|

+ O (x − c)2 ,

(|c|+2 sin c)

|c|2

2

(1.6a)

cos c−sin c)

+ 4(x − c) (|c|+2 sin c)(c

+ O (x − c)2 . (1.6b)

c|c|2

These series are unique, with successive terms telling us the values, gradients,

curvature, etc. of f and g around x = c.

If we attempt to expand about x = 0, however, we obtain two diﬀerent

series depending on whether we consider x > 0 and x < 0. The expansion of

1

1

g is g(x) = sign(x) − 3!

x|x| + 5!

x|x|3 − . . . , or to lowest order, just

g(x) = sign(x) + O x2 .

(1.7a)

Substituting this into f (x) = (1 + 2g(x))2 we have

f (x) = 5 + 4 sign(x) + O x2 .

(1.7b)

8

1 Origins of Discontinuity

This result is inconsistent, however, with the deﬁnition of f . Let us assume

that g lies between ±1 at the discontinuity, that is, −1 < g(0) < +1. Then

(1.7b) implies 1 < f (0) < 9. This is contrary to the deﬁnition of f in (1.11),

which reaches a minimum with respect to g at g = −1/2, where f = 0, and

therefore implies 0 < f (0) < 9.

We are only looking at behaviour at and near x = 0, so we should expect

the approximations of f and g to give consistent answers. The discrepancy

does not lie in the O x2 terms we have neglected, since they vanish for

small x. So what has gone wrong? How can we tell unambiguously the range

of values f takes as x changes sign and g jumps through the interval [−1, +1]?

The series expansions (1.6) to (1.7) are not strictly valid at x = 0 because

g and f are not continuous there, but there is a more useful way of looking

at what has gone wrong. The equation in (1.11) depends nonlinearly on the

discontinuous quantity g. In (1.7b) we are ignoring that nonlinearity, and

this, in fact, is the source of the contradictory ranges for f , not the series

expansion itself.

A better way to handle this turns out to be to deﬁne a switching multiplier

λ=

+1 if x > 0 ,

−1 if x < 0 ,

(1.8)

and to deﬁne this as lying in −1 < λ < +1 for x = 0. In terms of λ we can

write

g(x) = λ

sin x

x

and

2

f (x) = (1 + 2g(x)) ,

(1.9)

then expanding f gives

f (x) = 1 + 4g(x) + 4g(x)2

2

= 1 + 4λ sinx x + 4λ2 sinx2 x .

(1.10)

The term λ2 is simply unity for x = 0, and this is what went missing when

we approximated for small x above. If we are careful to keep the λ2 term, we

can now approximate for small x, and we obtain

f (x) = 1 + 4λ + 4λ2 + O x2 .

(1.11)

For −1 < λ < +1 the function g(0) = λ still takes values −1 < g(0) < +1,

but we now see that f correctly takes values 0 < f (0) < 9 (with a minimum

at λ = −1/2). Note how the ‘5’ from (1.7b) has become ‘1 + 4λ2 ’.

Why would it matter what values f passes through at the discontinuity?

One reason is that peaks or troughs—turning points with respect to λ—

in such a function can act like potential wells at the discontinuity, whose

presence or absence in a dynamic system may decide whether states can pass

through the discontinuity or become trapped within it.

1.3 Discontinuities in Approximations

9

Let us imagine that the discontinuity in f lies not perfectly at x = 0, but is

spread out over some |x| < ε, like the graphs shown on the left of Figure 1.5.

As we let ε tend to zero, we recover our discontinuous system, shown on the

right of Figure 1.5. Then consider a dynamical law

x˙ = −df /dx .

Figure 1.5 depicts three diﬀerent scenarios. If f is monotonic (top graph on

the left), then the variable x will evolve straight through the jump that occurs

at x ≈ 0. For some f with a peak or a trough around x = 0 (bottom two

graphs on the left), the variable x will get stuck in a potential well as it tries

to pass through the jump.

f

ε

ε

x

f

0

ε 0

ε

x

ε 0

f

x

f

ε

x

Fig. 1.5 A system ‘rolls’ down a potential φ, which has a jump over |x|

ε. In the limit

ε → 0, the shape of the potential at the jump becomes hidden inside the discontinuity.

In the limit ε → 0, these potential wells become squashed into the discontinuity at x = 0 and indistinguishable as a function of x (right-hand graph

in Figure 1.5). However, we can use nonlinear switching terms, as we used λ

in the graph of f above, to resolve the diﬀerence between the three cases.

What this exercise shows us is that:

• we can use switching multipliers like λ to endow discontinuities with nontrivial structure;

• we must respect nonlinear dependence on those multipliers.

Accepting that a system can depend nonlinearly on a discontinuous quantity essentially brings nonsmooth dynamics into the era of nonlinear switching

dynamics, into which this book is a ﬁrst tentative step. Already the outlook

appears to be as rich for nonsmooth systems as the era of nonlinear dynamics

has been for smooth systems. While this book seeks to set out the new tools

Hidden

Dynamics

The Mathematics of Switches,

Decisions and Other Discontinuous

Behaviour

Hidden Dynamics

Mike R. Jeffrey

Hidden Dynamics

The Mathematics of Switches, Decisions

and Other Discontinuous Behaviour

123

Mike R. Jeffrey

Department of Engineering Mathematics

University of Bristol

Bristol, UK

ISBN 978-3-030-02106-1

ISBN 978-3-030-02107-8 (eBook)

https://doi.org/10.1007/978-3-030-02107-8

Library of Congress Control Number: 2018959419

Mathematics Subject Classification: 00-02, 03H05, 34E10, 34E13, 34E15, 34N05, 37N25, 37M99,

41A60, 92B99, 70G60, 34C23

c Springer Nature Switzerland AG 2018

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of

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claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Arthur.

A smooth sea never made a skillful sailor

– African proverb

Life has no smooth road for any of us;

and in the bracing atmosphere of a high aim

the very roughness stimulates the climber to steadier steps. . .

– William C. Doane

Artwork by Martin Williamson (http://www.cobbybrook.co.uk/) 2017

Preface

Discontinuities are encountered when objects collide, when decisions are

made, when switches are turned on or oﬀ, when light and sound refract as

they pass between diﬀerent media, when cells divide, or when neurons are

activated; examples are to be found throughout the modern applications of

dynamical systems theory.

Mathematicians and physicists have long known about the importance of

discontinuities. Studying light caustics (the intense peaks that create rainbows or the bright ripples in sunlit swimming pools), George Gabriel Stokes

lamented to his ﬁanc´ee in a letter from 1857:

. . . sitting up til 3 o’clock in the morning

. . . I almost made myself ill, I could not get over it

. . . the discontinuity of arbitrary constants.

Discontinuities are not a welcome feature in dynamical or diﬀerential equations, because they introduce indeterminacy, the possibility of one problem

having many possible solutions, many possible behaviours. How interesting

it is then to consider the thoughts of the inﬂuential engineer Ove Arup:

Engineering is not a science . . . its problems are under-deﬁned,

there are many solutions, good, bad, or indiﬀerent.

The art is . . . to arrive at a good solution.

For Arup, ‘science studies particular events to ﬁnd general laws’. Many mathematical scientists would agree that the goal is to achieve generality and

banish indeterminacy. But why should the two be mutually exclusive?

Unlocking the potential of discontinuities requires tackling these issues of

determinacy and generality. While accepting that some parts of the world lie

beyond precise expression, discontinuities nonetheless give us a way to express

them, to approximate that which cannot be approximated by standard ‘wellposed’ equations, and to explore a world where certain things will ever remain

hidden from view.

ix

x

Preface

With these ideas in mind, this book attempts to ready the ﬁeld of nonsmooth dynamics for turning to a wider range of applications, simultaneously

moving beyond the traditional scope of, and bringing our subject closer into

line with, the traditional theory of diﬀerentiable dynamical systems.

At a discontinuity, we lose access to some of the most powerful theorems

of dynamical systems, and it has long been the task of nonsmooth dynamical

theory to redress this. Progress has been impressive in some areas, limited

in others. We suggest here that much of what has gone before constitutes

a linear approach to discontinuities, and here, we lay the foundations for a

nonlinear theory. Making use of advances in nonlinearity and asymptotics,

once we can extend elementary methods such as linearization and stability

analysis to nonsmooth systems, discontinuities stop being objects of nuisance

and start becoming versatile tools to apply to modelling the real world.

Several examples of applications are studied towards the end of the book,

and many more could have been included. Interest in piecewise-smooth systems has been spreading across scientiﬁc and engineering disciplines because

they oﬀer reliable models of all manner of abrupt switching processes. Our

aim is to set out in this book the basic methods required to gain an in-depth

understanding of discontinuities in dynamics, in whatever form they arise.

In this book, a discontinuity is blown up into a switching layer, inside

which switching multipliers evolve inﬁnitely fast across the discontinuity. Several concepts may be at least partly familiar in other areas of mathematics,

in particular algebraic geometry, boundary layers, singularity theory, perturbation theory, and multiple timescales. The terminology used here does not

exactly correspond to the usage in those ﬁelds, and attempting to refer to or

resolve all of the clashes in nomenclature would not make for an easier read.

Moreover, we do not use the concepts themselves in strictly the same way.

For example, we use the idea of an inﬁnitesimal ε-width of a discontinuity

that we can manipulate algebraically, but we are interested solely in the limit

ε = 0. This proves to be a suﬃciently rich problem, and though it raises the

question of what happens when we perturb to ε > 0, that is left for future

work. As we discuss in Chapter 1 and Chapter 12, more so than in any smooth

system, the perturbation of a discontinuity is a many faceted problem.

This work builds on the pioneering eﬀorts particularly of Aleksei Fedorovich Filippov, Vadim I. Utkin, Marco Antonio Teixeira, and Thomas

I. Seidman. I have been lucky to meet and work with all but the ﬁrst of

these, and much in the spirit of modern science, they represent the truly international and interdisciplinary endeavour of what was for too long a niche

ﬁeld of study.

In essence, the framework introduced in this book seeks to explore Filippov’s world more explicitly, to make non-uniqueness itself a useful modelling

tool in dynamics. Sitting somewhere between deterministic dynamics and

stochastic dynamics, nonsmooth dynamics oﬀers a third way: systems that

are only piecewise-deﬁned, rendering them almost everywhere deterministic.

Bristol, UK

Mike R. Jeﬀrey

Chapter Outline

The book is roughly split into three parts: introductory material in Chapters 1

and 2, fundamental concepts at the level of the student or non-expert in

Chapters 3 to 6 and Chapter 14, and advanced topics in Chapters 7 to 13.

Chapter 1 is almost a stand-alone and informal essay, surveying the reasons why discontinuities occur, what forms they take, why they matter, and

how imperfect our knowledge of them is. The chapter is intended to provoke

thought and discussion, not to be detailed reference on the many theoretical

and applied concepts it touches on.

Chapter 2 is a stand-alone “lecture”-style outline, a crash course on the

topic, and a taster of the main concepts that will be developed in the book.

Chapter 3 contains the complete foundation for everything that follows,

the formalities for how we deﬁne piecewise-smooth systems in a solvable

way. This chapter contains the elements necessary for the eager researcher

to rediscover for themselves the contents of the remainder of the book and

beyond.

Chapter 4 sets out the basic themes that dominate piecewise-smooth dynamics, the kinds of orbits, the key singularities, and the concepts of stability

and bifurcation theory.

Chapter 5 deﬁnes a general prototype expression for piecewise-smooth

vector ﬁelds in the form of a series expansion.

Chapter 6 describes the basic forms of contact between a ﬂow and a discontinuity threshold.

Chapter 7 contains the most important new theoretical elements of the

book, setting out the analytical methods required to understand piecewisesmooth systems.

Chapter 8 takes a step back, applying the previous chapters in the more

standard setting of linear switching.

Chapter 9 begins the leap forward into nonlinear switching, revealing some

of the novel phenomena of piecewise-smooth systems.

xi

xii

Chapter Outline

Chapter 10 focusses on the most extreme consequences of discontinuity,

via determinacy breaking and loss of uniqueness.

Chapter 11 tackles how we understand large-scale behaviour, with new

notions of global dynamics and associated bifurcations.

Chapter 12 asks how robust everything that has come before is. We consider how our mathematical framework can be interpreted in a practical setting and what happens to it in the face of nonideal perturbations.

Chapter 13 visits an old friend and long-term obsession of piecewisesmooth systems, the two fold singularity.

Chapter 14 is a series of case studies applying the foregoing analysis to

‘real-world’ models.

Exercises are provided at the back of the book to further facilitate a more

in-depth reading or lecture course.

How to Use This Book

This book will look rather diﬀerent to other works in the area. In Chapter 1,

we start from a tour of some less quoted, wide-ranging, but fundamental,

examples of how discontinuity arises. Chapter 3 presents the formalism for

studying nonsmooth dynamics that forms the foundation for everything that

follows and should be the starting point for any course. It is quite possible

to jump from there to Chapter 12 to focus on the application and robustness

of the formalism.

A proper understanding of the dynamics of nonsmooth system, or a course

in it, should progress through Chapters 4 to 11, and I would suggest focussing

on (and indeed extending) the analytical methods in Chapter 7. The great

peculiarities of nonsmooth systems begin to be revealed in Chapter 9 and

Chapter 10, and there are numerous examples therein to explore and build on.

More in-depth applications are given in the form of case studies in Chapter 14.

Exercises provided at the back of the book provide further insight into the

various examples and theorems explored, chapter by chapter.

Prerequisites. In reading this book, it will be helpful to have a grounding in (though we give elementary introductions where possible): single and

multi variable calculus, Taylor series, ordinary diﬀerential equations and elementary dynamical systems, some linear algebra (eigenvectors, etc.), and a

little introductory (highschool) physics. Applications will be explained with

background where they are discussed.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

1

Origins of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Discontinuities and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Discontinuities and Determinism . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Discontinuities in Approximations . . . . . . . . . . . . . . . . . . . . . . .

1.4 Discontinuities in Physics and Other Disciplines . . . . . . . . . . .

1.4.1 In Mechanics: Collisions and Contact Forces . . . . . . . .

1.4.2 In Optics: Illuminating a Victorian Discontinuity . . . .

1.4.3 In Sound: Wavefronts and Shocks . . . . . . . . . . . . . . . . . .

1.4.4 In Graphs: Sigmoid Transition Functions . . . . . . . . . . .

1.5 ‘Can’t We Just Smooth It?’ (And Other Fudges) . . . . . . . . . .

1.6 Discontinuities and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7 Discontinuity in Dynamics: A Brief History . . . . . . . . . . . . . . .

1.8 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

5

7

10

12

14

15

18

20

22

26

30

2

One

2.1

2.2

2.3

2.4

31

31

37

38

41

47

48

48

50

52

53

56

2.5

2.6

2.7

Switch in the Plane: A Primer . . . . . . . . . . . . . . . . . . . . . . .

The Elements of Piecewise-Smooth Dynamics . . . . . . . . . . . . .

The Value of sign(0): An Experiment . . . . . . . . . . . . . . . . . . . . .

Types of Dynamics: Sliding and Crossing . . . . . . . . . . . . . . . . .

The Switching Layer and Hidden Dynamics . . . . . . . . . . . . . . .

2.4.1 A Note on Modelling Basic Oscillators . . . . . . . . . . . . .

Local Singularities and Bifurcations . . . . . . . . . . . . . . . . . . . . . .

2.5.1 Equilibria and Local Stability . . . . . . . . . . . . . . . . . . . . .

2.5.2 Tangencies and Their Bifurcations . . . . . . . . . . . . . . . . .

2.5.3 Equilibria, Sliding Equilibria, and Their Bifurcations .

Global Bifurcations and Tangencies . . . . . . . . . . . . . . . . . . . . . .

Determinacy-Breaking: A First Glimpse . . . . . . . . . . . . . . . . . .

xiii

xiv

Contents

2.8

2.9

Counting Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3

The

3.1

3.2

3.3

3.4

3.5

3.6

Vector Field: Multipliers and Combinations . . . . . . . . . .

Piecewise-Smooth Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . .

The Discontinuity Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Constituent Fields and Indexing . . . . . . . . . . . . . . . . . . . . . . . . .

The Switching Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Inclusions and Existence of a Flow . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

61

62

63

67

69

72

4

The

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Flow: Types of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Types of Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Determinacy Breaking Events . . . . . . . . . . . . . . . . . . . . . . . . . . .

Equivalence and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Flow Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

73

74

79

83

88

89

90

5

The Vector Field Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 The Vector Field Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 The Canopy for One Switch . . . . . . . . . . . . . . . . . . . . . . .

5.1.2 The Canopy for Two Switches . . . . . . . . . . . . . . . . . . . . .

5.1.3 The Canopy for m Switches . . . . . . . . . . . . . . . . . . . . . . .

5.2 Deriving the Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Joint Expansions and Matching . . . . . . . . . . . . . . . . . . .

5.2.2 Series of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.3 Uniqueness of the Multilinear Term . . . . . . . . . . . . . . . .

5.3 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

91

92

93

94

95

95

96

98

100

6

Tangencies: The Shape of the Discontinuity Surface . . . . . .

6.1 Flow Tangencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Fold (d = 2, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Two-Fold (d = 3, k = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 Cusp (d = 3, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5 Swallowtail (d = 4, k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 Umbilic: Lips and Beaks (d = 4, k = 2) . . . . . . . . . . . . . . . . . . .

6.7 Fold-Cusp (d = 4, k = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8 Many-fold Singularities, Cusp-Cusps, and So On . . . . . . . . . . .

6.9 Proofs of Leading-Order Expressions for the Fold and

Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10 A Note on Alternative Classiﬁcations . . . . . . . . . . . . . . . . . . . . .

6.11 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

103

109

110

112

114

115

117

118

119

123

124

Contents

xv

7

Layer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 The Sliding Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.1 A Special Case: ‘Higher-Order’ Sliding Modes . . . . . . .

7.2 The Sliding Region’s Attractivity . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Singularities of the Sliding Manifold M . . . . . . . . . . . . . . . . . . .

7.4 End Points of the Sliding Region . . . . . . . . . . . . . . . . . . . . . . . .

7.5 Multiplicity and Attractivity of Sliding Modes . . . . . . . . . . . . .

7.5.1 One Switch, Multiple Sliding Modes . . . . . . . . . . . . . . .

7.5.2 Multiplicity of Sliding Modes at Intersections . . . . . . .

7.5.3 Classiﬁcation of Sliding Modes/Equilibria . . . . . . . . . . .

7.6 Layer Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.7 Equilibria and Sliding Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .

7.8 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9 Bifurcations of Equilibria and Sliding Equilibria . . . . . . . . . . .

7.10 A Saddlenode/Persistence Criterion . . . . . . . . . . . . . . . . . . . . . .

7.11 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

126

131

131

134

138

140

141

143

146

149

154

159

160

168

168

8

Linear Switching (Local Theory) . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1 The Sliding Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2 The Convex Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3 Equilibria and Sliding Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .

8.4 Boundary Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . . . .

8.4.1 One-Parameter BEBs in the Plane . . . . . . . . . . . . . . . . .

8.5 Boundaries of Sliding: For a Single Switch . . . . . . . . . . . . . . . .

8.5.1 Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.2 Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.3 Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.4 Swallowtail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.5 Umbilic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.6 Fold-Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6 Bifurcations of Sliding Boundaries in the Plane . . . . . . . . . . . .

8.7 Boundaries of Sliding: For r Switches . . . . . . . . . . . . . . . . . . . . .

8.8 The Hidden Degeneracy of Linear Switching . . . . . . . . . . . . . .

8.9 Piecewise-Smooth Time Rescaling . . . . . . . . . . . . . . . . . . . . . . .

8.10 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

172

173

174

175

175

180

182

183

185

186

188

190

192

195

198

199

200

9

Nonlinear Switching (Local Theory): The Phenomena of

Hidden Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 Nonlinear Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 Hidden Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.1 A Hidden van der Pol Oscillator . . . . . . . . . . . . . . . . . . .

9.2.2 Hidden Duﬃng Oscillator and Ueda chaos . . . . . . . . . .

9.2.3 Cross-Talk Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.4 Hidden Lorenz Attractor . . . . . . . . . . . . . . . . . . . . . . . . .

201

201

204

204

206

208

211

xvi

Contents

9.3

Hidden Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1 Cross or Not at an Intersection . . . . . . . . . . . . . . . . . . . .

The Illusion of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.1 Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.2 Slip Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.3 Switching with Time Dependence . . . . . . . . . . . . . . . . . .

Nonlinear Switching as a Small Perturbation . . . . . . . . . . . . . .

9.5.1 Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.2 Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.3 Structural Stability of the Sliding Manifold . . . . . . . . .

Hidden Degeneracy at Local Bifurcations . . . . . . . . . . . . . . . . .

9.6.1 Boundary Node Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.2 Fold-Fold and Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213

213

215

216

217

221

223

225

226

227

231

231

236

241

10 Breaking Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.1 Exit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.2 Exit Points: Deterministic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.2.1 Exit via a Simple Tangency . . . . . . . . . . . . . . . . . . . . . . .

10.2.2 Exit Transverse to an Intersection . . . . . . . . . . . . . . . . .

10.2.3 Exit via Tangency to an Intersection . . . . . . . . . . . . . . .

10.3 Exit Points: Determinacy-Breaking . . . . . . . . . . . . . . . . . . . . . .

10.3.1 Exit Transverse to an Intersection . . . . . . . . . . . . . . . . .

10.3.2 Exit via a Complex Tangency . . . . . . . . . . . . . . . . . . . . .

10.3.3 Zeno Exit from an Intersection . . . . . . . . . . . . . . . . . . . .

10.3.4 Exit from a Sliding Fold . . . . . . . . . . . . . . . . . . . . . . . . . .

10.4 Stranger Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.5 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243

243

244

244

245

247

250

251

257

263

269

270

272

11 Global Bifurcations and Explosions . . . . . . . . . . . . . . . . . . . . . . .

11.1 Local Classiﬁcation of Global Phenomena . . . . . . . . . . . . . . . . .

11.2 The Sliding Eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.3 Sliding Bifurcations/Explosions: The Global Picture . . . . . . .

11.4 Sliding Bifurcations/Explosions in Nonlinear Switching . . . . .

11.5 The Classiﬁcation and Its Completeness . . . . . . . . . . . . . . . . . .

11.5.1 Unfoldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.5.2 Classes of Sliding Bifurcation . . . . . . . . . . . . . . . . . . . . .

11.5.3 Classes of Sliding Explosion . . . . . . . . . . . . . . . . . . . . . .

11.5.4 The Omitted Singularities . . . . . . . . . . . . . . . . . . . . . . . .

11.6 Codimension Two Sliding Bifurcations and Explosions . . . . . .

11.7 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.8 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273

273

276

279

286

289

291

292

298

302

302

303

305

9.4

9.5

9.6

9.7

Contents

xvii

12 Asymptotics of Switching: Smoothing and Other

Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1 Probabilistic Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1.1 Multiplying Probabilities in the Combination . . . . . . .

12.1.2 The Unreasonable Eﬀectiveness of Nonsmooth

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.2 Convex Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.2.1 Experiments on Convex Switching . . . . . . . . . . . . . . . . .

12.2.2 Conclusion: Jitter Over the Convex Hull . . . . . . . . . . . .

12.3 Smooth Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.1 Why Smooth? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.2 The Smoothing Tautology . . . . . . . . . . . . . . . . . . . . . . . .

12.3.3 Deriving the Layer System via Smoothing . . . . . . . . . .

12.3.4 Equivalence of the Smoothed System? . . . . . . . . . . . . . .

12.3.5 Equivalence of Layer Dynamics . . . . . . . . . . . . . . . . . . . .

12.3.6 The Degeneracy of L Persists to L . . . . . . . . . . . . . . . .

12.3.7 Exponential Sensitivity, Contraction, and

Determinacy-Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.8 The Canopy as a Series Expansion . . . . . . . . . . . . . . . . .

12.4 Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.1 Extrinsic Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.2 Intrinsic Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.5 Intermediary Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.6 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

336

337

340

343

346

350

352

13 Four Obsessions of the Two-Fold Singularity . . . . . . . . . . . . . .

13.1 The Generic Two-Fold: A Summary . . . . . . . . . . . . . . . . . . . . . .

13.2 Obsession 1: The Prototype in n Dimensions . . . . . . . . . . . . . .

13.2.1 The Visible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.2.2 The Visible-Invisible Two-Fold . . . . . . . . . . . . . . . . . . . .

13.2.3 The Invisible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.2.4 Geometry of the Angular Jump Parameter ν + ν − . . . .

13.3 Obsession 2: The Nonsmooth Diabolo . . . . . . . . . . . . . . . . . . . .

13.3.1 First Return Map: The Skewed Reﬂection . . . . . . . . . .

13.3.2 The Rotation Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.3 Number of Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.4 The Nonsmooth Diabolo . . . . . . . . . . . . . . . . . . . . . . . . . .

13.3.5 The Nonsmooth Diabolo Bifurcation . . . . . . . . . . . . . . .

13.4 Obsession 3: The Folded Bridge . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.1 The Visible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.2 The Visible-Invisible Two-Fold . . . . . . . . . . . . . . . . . . . .

13.4.3 The Invisible Two-Fold . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.4.4 The Nonsmooth Diabolo Bifurcation: Sliding . . . . . . . .

355

356

359

361

362

363

365

367

367

369

371

378

380

385

387

388

389

390

307

307

309

311

314

314

320

321

321

322

328

329

329

335

xviii

Contents

13.5 Obsession 4: Sensitivity in the Layer . . . . . . . . . . . . . . . . . . . . .

13.5.1 The Unperturbed System . . . . . . . . . . . . . . . . . . . . . . . . .

13.5.2 The Perturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.6 An Unﬁnished Saga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

392

393

395

403

14 Applications from Physics, Biology, and Climate . . . . . . . . . .

14.1 In Control: Steering a Ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.2 Ocean Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3 Chaos in a Church . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.1 ‘Lumped Water’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.2 ‘Moving Point’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.3 ‘Discrete Kick’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.4 Explosion in a Superconducting Stripline Resonator . . . . . . . .

14.5 Conical Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.6 Optical Folded Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.7 Static Versus Kinetic Friction . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.8 A Paradox of Skipping Chalk . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.9 Pinching Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.9.1 Relaxation Oscillations and Canards . . . . . . . . . . . . . . .

14.9.2 Local Geometry of the Canard Singularity . . . . . . . . . .

14.9.3 The First Pinch: A Shot in the Dark . . . . . . . . . . . . . . .

14.9.4 The Second Pinch: Zooming in on the Manifolds . . . . .

14.9.5 The Third Pinch: Catching the Canards . . . . . . . . . . . .

14.10 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407

407

409

418

419

421

422

424

432

437

442

449

455

456

459

463

464

466

473

A

Discontinuity as an Asymptotic Phenomenon: Examples . .

A.1 Changes of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1.1 Large-Scale Bistability, Small-Scale Decay . . . . . . . . . .

A.1.2 Large-Scale Bistability, Small-Scale Dissipation . . . . . .

A.2 In Integrals: Stokes’ Phenomenon . . . . . . . . . . . . . . . . . . . . . . . .

A.3 In Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475

475

475

477

478

480

B

A Few Words from Filippov and Others, Moscow 1960 . . . 481

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Chapter 1

Origins of Discontinuity

Discontinuities occur when light refracts, when neurons or electronic switches

activate, and when collisions or decisions or mitosis or myriad other processes

enact a change of regime. We observe them in empirical laws, in the structure

of solid bodies, and also in the series expansions of certain mathematical

functions.

As commonplace as they may be, discontinuities are a curious thing to

try to build into dynamical models. They violate that central requirement of

calculus: to be continuous. They permit determinism to collapse in ﬂeeting

bursts of non-uniqueness. They conjure up a new realm of nonlinear dynamics.

In this ﬁrst chapter we start by exploring what it means for a system to be

discontinuous. Some discontinuities we understand, but many, the kinds of

discontinuities we encounter in engineering, the life sciences, and economics

and desperately wish to develop more sophisticated models for, we hardly

understand at all.

Formally, this book asks what happens to the trajectories of variables

x(t) = ( x1 (t), x2 (t), . . . , xn (t) ) ∈ Rn ,

(1.1)

as they evolve in time t, according to a set of ordinary diﬀerential equations

d

x = f (x) ,

dt

(1.2)

when the right-hand side is only piecewise-smooth, changing smoothly with

respect to x almost everywhere, except at certain thresholds σ(x) = 0 where

the value of f jumps, i.e. is discontinuous. But this rather dry statement hints

at few of the pitfalls and paradoxes of dynamics aﬄicted by discontinuities.

© Springer Nature Switzerland AG 2018

M. R. Jeﬀrey, Hidden Dynamics,

https://doi.org/10.1007/978-3-030-02107-8 1

1

2

1 Origins of Discontinuity

1.1 Discontinuities and Dynamics

When Isaac Newton set down the laws of motion that form the basis of

classical mechanics, he helpfully also set out the route to understand them

using calculus. Yet in doing so he mischievously threw into the stirring pot

some laws of motion not amenable to calculus. Century upon century since, a

juxtaposition of continuous and discontinuous change at the heart of physics

has remained, with consequences that remain only partly understood.

Collisions oﬀer a tangible example (Figu m

ure 1.1). Newton’s laws tell us the forces

v

n

acting on a moving object, and from those

lisio

M

-col

e

r

p

forces, calculus provides its speed and position. Yet when that object collides with another, instead of calculus we must employ a

M m

little mathematical sleight of hand. Calcun

isio

lus works for the pre-collision motion, and

coll

it works for the post-collision motion, but

then we must stitch the two together someu’

what artiﬁcially. To disguise the conceit—

m

n

v’ M

the discontinuity in the laws of motion—we

llisio

t-co

s

o

p

give the procedure a lofty title: an impact

law.

Fig. 1.1 Two objects collide and

Discontinuities allow us to gloss over

recoil. An impact law relates their

small details that seem to have no major

incoming speed u + v to their reeﬀect on our large-scale view. The last cencoil speed u + v by u + v =

e(u + v) for some 0 ≤ e ≤ 1.

tury, however, has taught us that no matter

how small, details can change everything.

The reason that we cannot follow motion through a collision, in the same

way we can follow objects that are rolling or in free ﬂight, is because the collision involves stepping between irreconcilable physical regimes: free motion

and rigid contact. One way to understand the regime change is to step into a

diﬀerent modelling approach entirely, perhaps on a ﬁner scale allowing bodies

to be more compliant and less idealized. But this can bring its own problems

and ambiguities, introducing much greater complexity, often probing areas

where our knowledge is less complete, and ultimately being diﬃcult to marry

up with the original discontinuous model.

To serve those situations, our task in this book, and in the ﬁeld of nonsmooth or piecewise-smooth dynamics more widely, is to provide a way within

a given dynamical model, to follow motion across the discontinuities between

irreconcilable regimes.

As science spreads its interest to new technological and sociological vistas,

it increasingly encounters a world full of irreconcilable regimes, of media not

behaving like steady waves rolling over the ocean, like electromagnetic waves

vibrating through spacetime, or like spheres orbiting and tumbling through

the vacuum of the heavens. Instead we ﬁnd abrupt changes that we patch

1.1 Discontinuities and Dynamics

3

reflectivity

over with ad hoc rules, such as switch

from behaviour A to behaviour B. Figure 1.2, for example, shows a discontinuity that turns up in climate models—

the reﬂectivity of the Earth’s surface

jumping across the edge of an ice shelf.

ce

The mathematical implications of such

tan

s

i

d

switches are not obvious.

Discontinuities like these are what

endow the world around us with struc- Fig. 1.2 A jump in surface reflectivity

between ice and water oceans.

ture. The boundaries of solid objects are

marked by jumps in properties like density, elasticity, or reﬂectivity. People

make decisions changing the course of their day. Storms and waves and glasses

break, social regimes change, lives are stopped and started.

As with collisions, we tend to skirt around the edges of these discontinuities

with a little sleight of hand and so describe almost everything going on in

a system, glancing over the discontinuities which, after all, are but ﬂeeting.

When I choose to go left or right, when a cell chooses to grow or divide, and

when a machine switches on or oﬀ—that brief moment when the choice in

enacted is trivial, isn’t it?

Far from it. Three centuries of calculus have left mathematicians uneasy

with discontinuities and reluctant to give up the continuity that provides

so many theorems concerning stability, attractors, bifurcations, and chaos,

because discontinuities leave these theorems in tatters.

From the mathematical point of view, a discontinuity renders a system

‘ill-posed’. A well-posed system has equations whose solutions: (i) exist, (ii)

are unique, and (iii) vary continuously with initial conditions. To satisfy all

three, a system must be smooth enough (meaning diﬀerentiable some number

of times, and certainly anything with a discontinuity does not qualify).

It turns out that at discontinuities we will often have to give up properties

(ii) and (iii), but not (i), not existence. It may seem perverse to give up

uniqueness and continuous dependence on initial conditions, but that is what

discontinuities are, events by which continuity and uniqueness are lost, and

our task is not to judge, but to learn how those losses can be exploited to

understand more about the world around us.

This book is an exploration of that idea. It is an attempt to extend

the methods of nonlinear dynamics beyond the barriers that discontinuities

have previously made impassable. In pushing back these boundaries, we ﬁnd

some intriguing behaviours. The methods, the theory behind them, and the

phenomena we discover, all require deeper future study. Though we prove

results where possible, not everything we do can be elevated to the level

of rigour that can be achieved with smooth systems (at least not yet), so

we do not claim a rigorous study here, only a development of ideas and

methods.

4

1 Origins of Discontinuity

Throughout the book we study the discontinuous system, with all of

the diﬃculties that brings, breaking only in Chapter 12 to consider nearby

‘perturbations’. There are various obvious ways that one may try to avoid

discontinuity depending on context. We might, for example, smooth out a

discontinuity, perhaps believing that smooth physical laws underlie it or

simply to make it easily computable. Or we might blur the discontinuity

with a distributive or stochastic process. An entire book mirroring this one

could be written using each approach, one smooth and deterministic and one

stochastic.

The discontinuous approach accepts that either of these, or numerous other

perturbations of the discontinuous model, could be the right approach. Let

us ﬁrst attempt to understand the underlying discontinuity, and later we will

probe a little into what happens when we perturb, in one way or another, by

smoothing, randomizing, or blurring the discontinuity in other ways.

The book starts and ends with less formal chapters which set the context

for our subject matter with the use of practical examples. This is one such

chapter and takes us on a short tour of how discontinuities arise and some

phenomena they produce. This expedition is not vital for those seeking an

introduction to piecewise-smooth dynamical systems theory, nor is it a comprehensive study of the topics touched on, but I hope you will at least skim

through it as motivation for what is to come.

In between those less formal chapters come more technical theory, aimed

at developing methods to understand the geometry and stability of solutions,

rather than focussing on proofs of solvability and universality of classes, but

opening numerous avenues for future study. After the theory is established

in Chapters 3 to 7 and explored at little in Chapters 8 to 11, we delve more

deeply into applications and ‘real-world’ switches in Chapters 12 to 14.

Towards the end of the book, we return to the question of what a discontinuity is. Discontinuities allow us to model abrupt change without imposing

undue structure. In a story that will unfurl as we reach Chapter 12, we will

learn that the best achievable representation of reality is not always the most

precise. We will see that it is sometimes unuseful, and even misleading, to

model processes in ﬁner detail than our understanding allows and that discontinuities provide not an obstacle to calculus but a new vehicle for it to

traverse uneven terrain.

To rely on continuity is to overlook that discontinuities are inescapable.

They arise not only in our everyday reality but within calculus itself, in the

midst of divergent series and singular perturbations, leaving mathematics no

less rich or rigorous for it. To rely on continuity is to risk overlooking that

diﬀerentiability reaches only so far into the complexities of a real world where

discordant media interact over disparate scales, and discontinuities are often

the result. We visit all of these in this chapter.

So let us see why discontinuity matters, where it comes from, and what it

looks like.

1.2 Discontinuities and Determinism

5

1.2 Discontinuities and Determinism

One issue will concern us only in limited situations, but will not go away

altogether, and that is:

where there are discontinuities there is non-uniqueness.

This non-uniqueness comes in many guises, but with just two main sources

that we can introduce brieﬂy.

The ﬁrst comes from a lack of knowledge of what happens inside a discontinuity. We may know that a quantity jumps between two values, but not

know precisely how it does so. We then use hidden terms to bring this uncertainty to life, to express the diﬀerent possible modes of behaviour inside the

jump. We shall show these constitute a form of nonlinearity. This is one of

the more subtle notions that will unfold throughout this book, and we will

introduce them a little more in Section 1.3.

The second source of non-uniqueness is more obvious, more well known,

and is the reason why mathematicians are taught a reluctance to study nonsmooth systems. It aﬄicts the solutions of a diﬀerential equation at a discontinuity. A classic example is the equation

dx

= |x|α ,

dt

(1.3)

for diﬀerent values of α ≥ 0. Its solutions take the form

x(t) = x0 1 +

1−α

t

x0 |x0 |−α

1/(1−α)

,

(1.4)

with an initial condition x(0) = x0 . Although we can write the solution

(fairly) simply, upon closer inspection we start to ﬁnd problems with it.

For α ≥ 1 solutions come in three types: those that start at x0 = 0 and sit

there forever, those that start at x0 < 0 and tend to x = 0 but never quite

reach it, and those that start at x0 > 0 and head oﬀ towards inﬁnity. For

instance, in the special case α = 1, we simply have dx

dt = |x|, and the solutions

become x(t) = x0 esign(x0 )t . The solution through any x0 is therefore unique: if

we know the ‘x0 ’ where we start, then all future (or indeed past) evolution of

x(t) is determined. This follows from the continuity of |x|α for α ≥ 1 (more

of

precisely the Lipschitz continuity of |x|α , by the so-called Picard-Lindel¨

theorem [149]).

For 0 < α < 1 the situation is entirely diﬀerent. The discontinuity in the

derivative of (1.3) takes over. Every solution through any x0 < 0 reaches

x = 0 in a future time t = |x0 |1−α /(1 − α), while every solution through any

x0 > 0 must have left x = 0 at a past time t = −|x0 |1−α /(1 − α). Does this

mean that we just have one solution that passes through zero? No, because

the point x = 0 is a solution itself. So if a solution from x < 0 reaches x = 0,

6

1 Origins of Discontinuity

it can sit there arbitrarily long before setting oﬀ again towards x > 0. This

means that an inﬁnity of diﬀerent solutions, all pausing to rest for diﬀerent

amounts of time at x = 0, all overlap at the origin and we cannot tell them

apart. As a result, the history and future of the point x0 = 0 are non-unique.

Non-unique histories are part of everyday experience and are one of the

reasons why nonsmooth systems have such broad applications. For example,

imagine an object that has been propelled along a surface and brought to rest

by friction. It is subsequently impossible to reconstruct the object’s motion

before it came to rest or to determine how much time has elapsed since it

stopped. A discontinuity in the frictional interaction between the object and

the surface has destroyed this information. This is an important eﬀect in our

everyday lives. When you hit the brakes in your car, you want them to behave

like 0 < α < 1 in the example above, to come to rest in ﬁnite time, not to

slow interminably towards the scene of an accident.

Non-unique futures are something less comfortable. A solution can start

out being unique and well behaved, but in the presence of a discontinuity, it

can ﬁnd itself ripped apart and endowed with inﬁnitely many possible futures.

We call these determinacy-breaking events.

Figure 1.3 depicts the scenario schematically. The picture shows the trajectories of a system evolving through space. Those trajectories are deterministic

everywhere except at a single point, the determinacy-breaking singularity.

Exit

trajectories

Inset

E I

determinacy

-breaking

Fig. 1.3 A determinacy-breaking event. Solutions before and after the singularity are

deterministic. Any trajectory starting in I hits the singularity. All trajectories in E originate

at the singularity. Inset right: forming a closed set.

Such singularities are common in nonsmooth systems. They result in new

kinds of nonlinear dynamics, new kinds of chaos and bifurcations, and even

new kinds of attractors. Imagine in Figure 1.3, for instance, if the inset I of

trajectories that are pulled into the singularity is intersected by the exit set

E of trajectories leaving the singularity (shown inset right). Then trajectories

will exist that make repeated yet unpredictable excursions, trapped forever

to return to the singularity, despite their exit path from it being uncertain.

With its inherent ambiguities of various sources of non-uniqueness, it is

easy to dismiss discontinuities from serious dynamical theory. But the nonuniqueness turns out to be useful, not to be swept under the rug or axiomatized into oblivion, and closely intwined in all its forms with nonlinearity.

1.3 Discontinuities in Approximations

7

1.3 Discontinuities in Approximations

How do you approximate near a discontinuity? This is what we are doing

very often when we are studying discontinuous systems and their dynamics, whether in theoretical equations or in empirical models. Consider the

following.

Example 1.1 (Approximating a Nonlinear Switch). Let us try to approximate

a pair of functions

g(x) =

sin x

|x|

and

2

f (x) = (1 + 2g(x)) ,

(1.5)

sketched in Figure 1.4. (In a strict sense we should not refer to f and g as

functions if they take many values at x = 0, but we allow this small abuse

of terminology, much as the Heaviside step ‘function’ or sign ‘function’ are

so-called, with the values at x = 0 being, after all, our topic of interest).

+1

9

g(x)

0

f(x)

−1

0

x

1

0

0

x

Fig. 1.4 The graph of two functions g(x) and f (x) with a discontinuity at x = 0.

These are both well behaved for x away from zero, and if we wish to approximate them near a point c = 0, we can expand them as Taylor series,

g(x) =

f (x) =

sin c

|c|

c−sin c

+ (x − c) c cosc|c|

+ O (x − c)2 ,

(|c|+2 sin c)

|c|2

2

(1.6a)

cos c−sin c)

+ 4(x − c) (|c|+2 sin c)(c

+ O (x − c)2 . (1.6b)

c|c|2

These series are unique, with successive terms telling us the values, gradients,

curvature, etc. of f and g around x = c.

If we attempt to expand about x = 0, however, we obtain two diﬀerent

series depending on whether we consider x > 0 and x < 0. The expansion of

1

1

g is g(x) = sign(x) − 3!

x|x| + 5!

x|x|3 − . . . , or to lowest order, just

g(x) = sign(x) + O x2 .

(1.7a)

Substituting this into f (x) = (1 + 2g(x))2 we have

f (x) = 5 + 4 sign(x) + O x2 .

(1.7b)

8

1 Origins of Discontinuity

This result is inconsistent, however, with the deﬁnition of f . Let us assume

that g lies between ±1 at the discontinuity, that is, −1 < g(0) < +1. Then

(1.7b) implies 1 < f (0) < 9. This is contrary to the deﬁnition of f in (1.11),

which reaches a minimum with respect to g at g = −1/2, where f = 0, and

therefore implies 0 < f (0) < 9.

We are only looking at behaviour at and near x = 0, so we should expect

the approximations of f and g to give consistent answers. The discrepancy

does not lie in the O x2 terms we have neglected, since they vanish for

small x. So what has gone wrong? How can we tell unambiguously the range

of values f takes as x changes sign and g jumps through the interval [−1, +1]?

The series expansions (1.6) to (1.7) are not strictly valid at x = 0 because

g and f are not continuous there, but there is a more useful way of looking

at what has gone wrong. The equation in (1.11) depends nonlinearly on the

discontinuous quantity g. In (1.7b) we are ignoring that nonlinearity, and

this, in fact, is the source of the contradictory ranges for f , not the series

expansion itself.

A better way to handle this turns out to be to deﬁne a switching multiplier

λ=

+1 if x > 0 ,

−1 if x < 0 ,

(1.8)

and to deﬁne this as lying in −1 < λ < +1 for x = 0. In terms of λ we can

write

g(x) = λ

sin x

x

and

2

f (x) = (1 + 2g(x)) ,

(1.9)

then expanding f gives

f (x) = 1 + 4g(x) + 4g(x)2

2

= 1 + 4λ sinx x + 4λ2 sinx2 x .

(1.10)

The term λ2 is simply unity for x = 0, and this is what went missing when

we approximated for small x above. If we are careful to keep the λ2 term, we

can now approximate for small x, and we obtain

f (x) = 1 + 4λ + 4λ2 + O x2 .

(1.11)

For −1 < λ < +1 the function g(0) = λ still takes values −1 < g(0) < +1,

but we now see that f correctly takes values 0 < f (0) < 9 (with a minimum

at λ = −1/2). Note how the ‘5’ from (1.7b) has become ‘1 + 4λ2 ’.

Why would it matter what values f passes through at the discontinuity?

One reason is that peaks or troughs—turning points with respect to λ—

in such a function can act like potential wells at the discontinuity, whose

presence or absence in a dynamic system may decide whether states can pass

through the discontinuity or become trapped within it.

1.3 Discontinuities in Approximations

9

Let us imagine that the discontinuity in f lies not perfectly at x = 0, but is

spread out over some |x| < ε, like the graphs shown on the left of Figure 1.5.

As we let ε tend to zero, we recover our discontinuous system, shown on the

right of Figure 1.5. Then consider a dynamical law

x˙ = −df /dx .

Figure 1.5 depicts three diﬀerent scenarios. If f is monotonic (top graph on

the left), then the variable x will evolve straight through the jump that occurs

at x ≈ 0. For some f with a peak or a trough around x = 0 (bottom two

graphs on the left), the variable x will get stuck in a potential well as it tries

to pass through the jump.

f

ε

ε

x

f

0

ε 0

ε

x

ε 0

f

x

f

ε

x

Fig. 1.5 A system ‘rolls’ down a potential φ, which has a jump over |x|

ε. In the limit

ε → 0, the shape of the potential at the jump becomes hidden inside the discontinuity.

In the limit ε → 0, these potential wells become squashed into the discontinuity at x = 0 and indistinguishable as a function of x (right-hand graph

in Figure 1.5). However, we can use nonlinear switching terms, as we used λ

in the graph of f above, to resolve the diﬀerence between the three cases.

What this exercise shows us is that:

• we can use switching multipliers like λ to endow discontinuities with nontrivial structure;

• we must respect nonlinear dependence on those multipliers.

Accepting that a system can depend nonlinearly on a discontinuous quantity essentially brings nonsmooth dynamics into the era of nonlinear switching

dynamics, into which this book is a ﬁrst tentative step. Already the outlook

appears to be as rich for nonsmooth systems as the era of nonlinear dynamics

has been for smooth systems. While this book seeks to set out the new tools

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