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Hidden dynamics the mathematics of switches, decisions and other discontinuous behaviour

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


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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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