Hidden dynamics the mathematics of switches, decisions and other discontinuous behaviour
Mike R. Jeffrey
The Mathematics of Switches, Decisions and Other Discontinuous Behaviour
Mike R. Jeffrey
Hidden Dynamics The Mathematics of Switches, Decisions and Other Discontinuous Behaviour
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 the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional 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
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
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
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
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
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.
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 ,
as they evolve in time t, according to a set of ordinary diﬀerential equations d x = f (x) , dt
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.
M. R. Jeﬀrey, Hidden Dynamics, https://doi.org/10.1007/978-3-030-02107-8 1
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
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.
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
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
for diﬀerent values of α ≥ 0. Its solutions take the form x(t) = x0 1 +
1−α t x0 |x0 |−α
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 ). 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,
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.
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
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|
f (x) = (1 + 2g(x)) ,
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).
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
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 .
Substituting this into f (x) = (1 + 2g(x))2 we have f (x) = 5 + 4 sign(x) + O x2 .
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 ,
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
f (x) = (1 + 2g(x)) ,
then expanding f gives f (x) = 1 + 4g(x) + 4g(x)2 2
= 1 + 4λ sinx x + 4λ2 sinx2 x .
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 .
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
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.
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