This paper represents the author’s view on the impact of the book *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields* by John Guckenheimer and Philip Holmes, first published in 1983 (Springer-Verlag, Berlin). In particular, the questions addressed are: if one were to write a similar book for the 21st century, which topics should be contained and what form should the book take in order to have a similar impact on the modern generation of young researchers in applied dynamical systems?

## Introduction

When I was a graduate student in the late 1980s, there was one textbook that was the greatest source of wisdom to almost all the researchers I came into contact with. This was the book *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields*, by Guckenheimer and Holmes (1). Although today there are hundreds of graduate texts with at least some of these words (and almost certainly, in addition, “chaos”) in the title; then, there were very few. There were the pure maths books—somewhat specialized and lacking the big picture—and a few engineering primers, which viewed the nonlinear world as through a glass dimly. But Guckenheimer and Holmes' book was the real McCoy, solid nourishment. It was quoted by each week’s seminar speaker, much thumbed in libraries, discussed in reading groups, and always referenced in theses by Ph.Ds who had bought their own copy, decorated it with highlighter pens and coffee, and had even taken it home as bedtime reading.

Today, the dynamical systems research community is far more mature, and the nomenclature of bifurcation theory is hardwired into almost all fields of scientific study. But, cast your mind back to 1983; chaos was all the rage, like David Bowie’s latest haircut, and it seemed that “chaos theory” could be just another passing fad like catastrophe theory, cold fusion, or superstring theory. In one sense it was (to quote Holmes (2)): “In spite of all the hype and my enthusiasm for the area, I do not believe that *chaos theory* exists, at least not in the manner of quantum theory … I therefore prefer a sober description of new tools [many of them from the classical theory differential equations, and a guiding global geometrical viewpoint that originated with Poincaré over a hundred years ago], rather than grand claims that the problems of life, the universe, and everything will shortly be solved.”

Nevertheless, back in 1983 “nonlinearity” was still treated as an aberration, a pathology to be avoided; an artifact, not the effect. That nonlinear dynamics was something one could handle, unravel, and, to a large extent, exploit, was a realization that swept rapidly across campuses the world over throughout the 1980s. This revolution came about not in the least because of the way that Guckenheimer’ and Holmes’ book (1) made the esoteric world of phase space topology, bifurcation, homoclinics, horseshoes, and strange attractors accessible to all numerate researchers. Moreover, through center manifolds, normal forms, symbolic dynamics, and the Mel’nikov method, we were given the ability to apply this theory not just to hypothetical butterfly attractors and tent maps, but to real-world applications.

Almost as if a sacred text, the book (1) is eclectic; it contains historical narrative, self-help “how-to” sections, and plenty of examples of how to interpret the scripture. It also looks to the future, with countless open questions (some of which, we are warned would form suitable Ph.D. topics). There is a strong flavor of applied mechanics, from which discipline most of the examples are taken—hence, the these-days unfashionable term *nonlinear oscillations* in the title. The book is dedicated “to G. Duffing, E. N. Lorenz and B. van der Pol, pioneers in a chaotic land,” and indeed, the models that bear their names form three of the four motivating account. If van der Pol, Lorenz, and Duffing were the founding fathers who spoke of the promised land; then, Guckenheimer and Holmes’s achievement was to lay down the ground rules for the early settlers. Moreover, a measure of their success is that the promised land is now a complex society. Nonlinear dynamics pervades economics, biology, psychology, physics, chemistry, medicine, etc.; in fact, almost all fields of academic endeavor. “Nonlinear science is like nonelephant zoology,” famously stated Stanislaw Ulam.

There was never a second edition (although there were reprintings with updated bibliographies and extensive corrections, e.g., (3)). The question then is, what would need to be added to the book to make it reflect the advances in applied dynamical systems that have occurred in the intervening 20 years or so. What would a modern-day guidebook on applied dynamical systems contain? Given the depth and strength of our current knowledge, perhaps it might be better to liken what is required to a modern-day (Alexandria) library rather than an updated single tome. Nevertheless, given the amount of material that is out there, there is arguably an even more pressing need for a single expository text that is at places broad brush and informal, at other places detailed and didactic, featuring real-world case-study examples, yet full of open questions, always leaving the reader wanting more—just as Guckenheimer and Holmes’ book (1) did more than 20 years ago. Of course, there could be many possible such books, reflecting the diverse world views of the modern dynamical systems research community. Nevertheless, this paper shall attempt to spell out, in three separate sections, what should be covered from the personal perspective of this author.

## Mathematical Advances

My thesis advisor once told me: “If you are worried that your new Theorem in dynamical systems was already proved 30 years ago by some obscure Russian, then stop worrying: all Theorems in dynamical systems were proved 30 years ago by some obscure Russian!”

Prior to 1983, this was possibly sound advice. But the book (1) went a long way to rectifying our ignorance in the West. For example, instead of “period-three implies chaos” (4), we have the full statement of Ŝarkovskii’s theorem (5) of which Li and Yorke’s (4) result is a special case. In addition to the great hero of the West, Steve Smale, the Russian heroes Lyapunov, Andronov, etc., get their just desserts. The book also introduces, via the theory of averaging, Mel’nikov’s method for analyzing homoclinic chaos in maps, and there is a key section on homoclinic orbits in flows that pays due credit to L. P. Shil’nikov.

There have since been other books that expand on these ideas. Wiggins (6,7) gives further developments of Melnikov’s method and how to apply it to prove the existence of chaotic dynamics. Kuznetsov (8) gives, not only an encyclopedic tour of bifurcations of codimension-one and two, but also greatly refines the tools for analyzing them. He shows how to compute normal form coefficients by projection and how this can be done in a global way without first putting the linear part into canonical form. Shil’nikov et al. (9,10) have added further rigor and geometric techniques for analyzing global bifurcations. Expanding on these developments, our twenty-first century book should probably also say something on the Lin-Sandstede method, an analytic technique for analyzing the dynamics near homoclinic and heteroclinic orbits (11,12). For example, just like period-doubling cascades, there are certain situations that lead to *homoclinic doubling cascades* (13).

There is also now a much better understanding of strange attractors themselves than there was in 1983. Ergodic theory has led to further ways of characterizing them. There are also explanations of *boundary crises* (14) that lead to their creation or destruction, and theories of *riddled basins of attraction* (15) for attractors that lie on invariant manifolds. In flows with quasiperiodic forcing, there are also theories suggesting the existence of strange nonchaotic attractors (16), although whether or not they truely exist remains an open question. Indeed, the sequence of metamorphoses that occur as an invariant torus loses its smoothness and ceases to exist has many subtleties that are not widely understood (but see, e.g., (17,18 19)). A text that presents a clear pedagogical path through this literature is pressing and should form part of our new book.

The book (1) deals with only two kinds of dynamical system: maps and flows in finite dimensions. There are now considerable bodies of literature on infinite dimensional dynamical systems, arising from evolution partial differential equations (20), from delay-differential equations (21) or from lattice ordinary differential equations, see, e.g., (22,23). Many fundamental existence results have now been established, and it is high time, it would seem for a twenty-first century book to “do a Guckenheimer and Holmes” on this material (but see, e.g., the book by Robinson (24) for the first steps in this direction). There is, of course, a huge amount to infinite-dimensional dynamical systems theory, with many modern applications in for example optics and information storage, (25) complex networks (26) and descriptions of turbulence (27) (…a term that in the early days would be banded around almost synonymously with “chaos,” e.g., (28)). Further subclassification of the many exciting directions within infinite dimensional systems, with their strong links to functional analysis, would require more than a single book. Thus, for brevity, while providing pointers to the infinite-dimensional literature, let us suppose our new book concerns itself exclusively with finite-dimensional systems.

Since the 1980s, there has been an explosion of interest in flows with additional structure, notably symmetry. Early results in this area (29,30) took a more analytical approach to bifurcation, relying on the implicit function theorem and Lyapunov-Schmidt reduction. More recently, a geometric theory of systems with symmetry as in (1) has begun to emerge, which will clearly need to be covered in our twenty-first century book. Various intricacies involve isotropy subgroups and the equivalent branching lemma, mode interactions, ponies on a merry-go-round, spatiotemporal complexity, robust heteroclinic bifurcations, and the dynamics of coupled cell networks; see, e.g., (31) for a selection of recent results.

Hamiltonian systems have received much attention in their own right because of their relation to physics and mechanics. Many recent themes can be gleaned from the recent volumes in honor of the 60th birthdays of Arnol’d (32) and Marsden (33). Clearly our twenty-first century guidebook cannot do justice to this literature that has run in parallel with that on dissipative dynamical systems. However, it may be pertinent to show how Hamiltonian systems relate to other systems with structure, such as reversible systems (34) and those which merely preserve a first integral.

Another interesting direction has been the dynamics of nonsmooth systems. As we shall see in Sec. 4, such problems readily arise in mechanics as well as through the increasingly important concept of hybrid systems that contain both an analog and a digital component (35). An interesting research direction that requires a systematic treatment (but, see (36,37)) is a classification of nonsmooth systems by means of unfolding nongeneric interaction with surfaces in phase space that separate smooth motions. Here the *discontinuity mapping* idea due to Nordmark (38), see also (39), provides a particular method for deriving “normal form” maps with jumps (in a certain derivative). Closely related to such systems are differential algebraic problems, where one has *singularity induced bifurcations*. Other emerging classes of dynamical systems are noisy and uncertain systems, as well as systems or attractors that are only defined for a finite time. Some of these directions are discussed in (40), but perhaps the final chapter of our new book should provide an attempt to put these emerging classes of dynamical system into context.

## Numerical Methods

“Moore’s law states that computing power doubles every 1.5 years.”

Not only has computer hardware improved beyond all recognition, but since the 1980s ever more sophisticated computational methods have been derived for analyzing dynamical systems. At the same time, there have been advances in numerical analysis, which have increased our confidence in the faithfulness of these methods. These developments mean that numerics can no longer be seen as a blunt instrument to be turned to only when all analytical methods fail, but can be used as a precise instrument for probing structures in phase and parameter space.

In terms of pure simulation, it has been shown that many numerical schemes (such as Runge-Kutta methods) whose classical error estimates grow like $ekT$, where $T$ is the computation time, do, nevertheless, faithfully reproduce bounded attractors and other phase space structures as $T\u2192\u221e$ (see, e.g., the book by Humphries and Stuart (41)). Such analyses rely on treating the numerical method as a discrete-time dynamical system that approximates the true continuous-time system. This approach has spawned a whole new methodology, that of *geometric integration* (42 43), which derives methods that specifically preserve certain of the properties of the underlying dynamical system, such as Hamiltonian structure or conserved quantities.

In addition, direct numerical methods have been defined for computing invariant sets and their stable and unstable manifolds. These methods are often implemented in a numerical path-following environment, where the set in question is continued in a bifurcation parameter; see, e.g., (44). Moreover, techniques exist for automatic detection of local bifurcation points, unfolding the ensuing bifurcation, branch-switching to the bifurcating solutions, and continuation of the bifurcation curve in two or more parameters. These methods come in various software implementations, for example the codes AUTO, DSTOOL, MATCONT and LOCA details of many of which are available from the DSWeb site (45).

In the 1980s, there was the fantastic book by Abrahams and Shaw (46) that used artistic techniques to represent phase space structures and bifurcation diagrams in three dimensions. We now have the tools available to view such structures using computer animation, using clever numerical algorithms that compute invariant manifolds and attractors directly; see, e.g., (47,48). The unstable manifold of the Lorenz equations has even been rendered in crochet (49). Our new twenty-first century text should almost certainly contain, as a Web supplement, a modern-day computer-animated version of Abrahams and Shaw (1) that uses no artistic license, just solutions of ordinary differential equations (ODEs), to plot the complex geometric structures. Moreover, countless web applets now exist (for example, just type “Lorenz attractor” into Google and see what you get) that allow one to simulate dynamical systems and explore their phase space at the click of a mouse. Perhaps the user should be invited to create their own mathematical artwork in this way. Not so much as a guidebook on dynamical systems, but an online oracle. Submit your system. Tell the oracle to compute its attractors, spot interesting parameter values, detect bifurcations, draw the relevant pictures, write the paper, give the conference presentation for you, … perhaps not.

## Applied Mechanics

Q. What is the difference between a physicist, a mathematician, and an engineer?

A. For a physicst, it is most important to be the first. For a mathematician, it is instead paramount to be right. Yet, for the engineer, it is most vital to deliver on Friday!

It is a fair question to ask, what has been the *impact* of our more refined view of dynamical systems since 1983. Has nonlinear dynamics and chaos truly *delivered*?

Of course, a true answer to such a question should look at many different nonelephant scientific fields of endeavor. Indeed, one could hardly fail to note the significant advances that have occurred in statistics and data analysis, e.g., (50), in networks and communications, e.g., (51), in biology and medicine, e.g., (52), in economics and the social sciences, e.g., (53), and, indeed, in abstract areas of mathematics, such as number theory, quantum mechanics, and functional analysis, e.g., (54). Yet, in keeping with the flavor of (1), let us suppose that applications in our new book are restricted to applied mechanics.

Much of the elementary theory of dynamical systems can be explained in terms of pendulum oscillations (see, e.g., the primer (55) that uses just such a paradigm to explain nonlinear physics). Indeed, the forced Duffing oscillator example in (1) is just a truncation of the full sinusoidal nonlinearity. Such nonlinear motion has been exploited in tuned, nonlinear vibration absorbers (56) using the principle of autoparametric resonance. Such ideas are having a resurgence in micro-electromechanical systems (MEMS) (57). Here, nonlinear forces, such as electrostatics, squeeze film damping and large deflection effects that are neglectable at macroscales can become the dominant terms for devices of very low mass that operate in near vacuum conditions. See, e.g., (58,59 60), for recent applications.

Dynamical systems theory has also proved useful in describing nonlinear statics, specifically in long quasi-one-dimensional structures (rods, beams, struts, etc.), where an (approximated as infinite) space coordinate plays the role of time. Mel’nikov theory has been used to predict *spatial chaos* (61). More crucially, though, if the original buckling (pitchfork) bifurcation is subcritical, such structures have the propensity to localize their deformation, corresponding to a homoclinic orbit in space (62). Application of these ideas has been used to explain writhing of DNA strands, buckling of undersea cables, and, in a beautiful piece of work by McMillen and Goriely (63), the *perversion* observed in the tendrils of climbing plants. Closely related to these ideas is the “nucleation” of localized pattern in space using, for example, fourth-order Swift-Hohenberg theory (64). It is pressing for the theory behind these examples to be expounded in its simplest possible form and put into context.

The fourth motivating example in (1) is that of a ball bouncing on a perfect table. This is a canonical impact oscillator, which was also further elucidated in the near contemporaneous work of Shaw and Holmes (65) (although there were earlier related studies by Peterka, see (66), and others in the East). Vibroimpacting dynamics remain one of the most fundamental problems in engineering mechanics. Often impacts and other nonsmooth events represent the grossest form of nonlinearity in engineered systems, yet it is the hardest to understand. Impact oscillators are at the heart of how clocks work, how bang-bang control works, and how power electricians convert from one DC voltage to another; see, e.g., (36). The book could mention one or two such applications where dynamical systems analysis has made a difference. One such problem is limit-cycle oscillations caused by freeplay in aircraft wingflaps (67), another is an understanding of percussive drilling and moling (68).

An increasingly important strand of research is biomechanics. Early work of Holmes considered the mechanics of swimming fish (69), a theme of his research that continues (70). More recently, he has written extensively on the mechanics of both human and insect walking (71). For example, the basic underlying mechanism to the human gait (72) would appear to be an inverted pendulum (in stance phase) that lifts off and undergoes free (resonant?) motion up to the point where an impact occurs. Limps are period-doubled motions. Moreover, the transition from running to walking can be viewed as a nonsmooth bifurcation upon variation of key parameters, such as the desired forward speed. Such models will provide suitable terrain for our twenty-first century book and can be compared to experiments and anatomical data.

Biomechanics is but one example of the growing interface between dynamical systems and the life sciences, which is clearly going to be of increasing importance as we understand more of the complex processes that occur within and between organisms and groups of organism. At the bottom level, we are driven by biochemistry—the law of mass action, which is fundamentally nonlinear. Gene expression (so-called transcriptiomics and proteomics) represent complex networks of interactions, for which we have increasing databanks through bioinformatics. Then cells signal via excitable mechanisms, they flow through diffusion gradients, they communicate through nerves, endocrinology, etc. (52). And so-on up through the length scales. There is much for us to do, and a new science is emerging: *systems biology* (73) (a term that sits rather like “chaos theory” did in 1983—only time will show whether it persists).

## Conclusion

If physics was the great science of the twentieth century, then surely it will be the life sciences that increasingly dominate in the twenty-first century, to say nothing of the social sciences: economics, models of human behavior, perception, and cognition. All of these will need the insight of applied dynamical systems theory, and the way of working that was pioneered by Guckenheim and Holmes book (1) all those years ago. This author certainly has no time, energy, nor ability to write the updated volume that is required to address these future challenges. In fact, where is the motivation in our current grant-getting, league-table-infested, financially astute university culture for any of us to write such a key text? Maybe Guckenheim and Holmes should be encouraged so to do.

## Afterword: Meeting the Author

Toward the end of my time as a graduate student, in January 1991, I had the opportunity to travel to my first international conference. Dynamics Days, Houston. The first time I had ever crossed the Atlantic. After delays and a fog-bound unscheduled overnight stop in New Orleans, I arrived at 10:00 A.M. on the first day of the meeting, to find that my poster slot was in $15min$. Badly jet-lagged, I pinned my dismal pieces of A4 paper to the posterboard and went in search of some strong coffee (not an easy feat in Texas). On my return, someone was reading my poster: “*Could you please explain*…?” “Okay,” I said, “Do you know what a normal form is?” “Have you seen the work by Sethna and Shaw…?” “*Sure I have. Steve Shaw was my student!*” And only then did I think to look at his badge to find out with whom I was talking… . Oh, and he had a gray beard!

## Acknowledgment

This paper is dedicated to Phil Holmes on the occasion of his 60th birthday. I should like to thank Oliver O’Reilly, Harry Dankowicz, and Bernd Krauskopf for their comments on an earlier draft.

*The essence of chaos*,' by E. N. Lorenz