# History of dynamical systems

Post-publication activity

Curator: Philip Holmes

Dynamical systems theory (also known as nonlinear dynamics, chaos theory) comprises methods for analyzing differential equations and iterated mappings. It is a mathematical theory that draws on analysis, geometry, and topology – areas which in turn had their origins in Newtonian mechanics – and so should perhaps be viewed as a natural development within mathematics, rather than the scientific revolution or paradigm shift that some popular accounts have suggested. (The fact that a given deterministic dynamical system can be proven to possess chaotic (or stable) solutions does not necessarily imply that the phenomenon that it purports to describe behaves likewise. That will depend on the quality of the mathematical model.) However, see Aubin and Dahan Dalmedico (2002) for a 'sociohistorial' analysis which discusses extra-mathematical influences and describes the confluence of ideas and traditions that occurred in the turbulent decade around 1970.

This article provides a brief, and perhaps idiosyncratic, introductory review of the early history of the subject, from approximately 1885 through 1965. While I take a mathematical viewpoint, I do not intend to downplay the important motivations and contributions to the field from outside mathematics per se, and especially from the physical sciences. I include some references to this work, but to do it justice would require a separate article.

In compiling the bibliography I have emphasized original references and a few relatively early review articles. I discuss only deterministic systems; there is also a growing qualitative theory of stochastic dynamical systems, see, e.g. L. Arnold (1998). The important topic of ergodic theory (Katok and Hasselblatt, 1995) is mentioned only in passing.

## Poincaré and Birkhoff

The qualitative theory of dynamical systems originated in Poincaré's work on celestial mechanics (Poincaré 1899), and specifically in a 270-page, prize-winning, and initially flawed paper (Poincaré 1890). The methods developed therein laid the basis for the local and global analysis of nonlinear differential equations, including the use of first-return (Poincaré) maps, stability theory for fixed points and periodic orbits, stable and unstable manifolds, and the Poincaré recurrence theorem. More strikingly, using the example of a periodically-perturbed pendulum, Poincaré showed that mechanical systems with $$n \ge 2$$ degrees of freedom may not be integrable, due to the presence of homoclinic orbits. The methods that he developed were subsequently described in detail in the three-volume treatise Poincaré (1892-1899).

Following this and Hadamard's studies of geodesic flows, G.D. Birkhoff showed that, near any homoclinic point of a two-dimensional map, there is an infinite sequence of periodic orbits whose periods approach infinity (Birkhoff 1927), and that annulus maps having orbits with different periods can possess complicated limit sets separating their domains of attraction (Birkhoff 1932), of which more below.

## Andronov, Pontryagin and the Moscow School

During the period that Birkhoff was working in the U.S., A.A. Andronov (1901-1952), a student of L. Mandelstam (1879-1944), established a strong group in dynamics in the U.S.S.R at Gorki (Nizhni-Novgorod). (For more on Andronov's work in control theory and the Mandelstam school in mathematical physics, see http://ict.open.ac.uk/reports/1.pdf.) Andronov and Pontryagin (1937) introduced structural stability (systèmes grossiers, or coarse systems) and began a study of local bifurcations that was continued beyond Andronov's premature death by his wife, Leontovich (Andronov et al 1971, 1973). Andronov had trained as a physicist, and unlike much of the work in Europe and the U.S., his group remained close to applications, especially in nonlinear oscillations and waves. Indeed, Andronov, Vitt, and Khaiken (1966), an expanded version of a book originally published in 1937, remains an excellent introduction to nonlinear systems for applied scientists. Subsequently M. Peixoto (originally trained as an engineer) generalised Andronov and Pontryagin's theory of planar systems to two-dimensional manifolds (Peixoto 1962).

Figure 2: Andrey Nikolaevich Kolmogorov in the 1940s.

A more abstract approach, developed in Moscow, gained attention outside the U.S.S.R. via the translation of (Nemytskii and Stepanov, 1960), introduced by S. Lefschetz, who had himself published a text on qualitative theory a few years earlier (Lefschetz 1957). Here the first clearly-defined strange attractor – the solenoid – was described. The works of Kolmogorov, Anosov, Arnold and Sinai grew out of this "Moscow school" in the 1950-60's, with important work on ergodic theory (Sinai, 1966), geodesic flows (Anosov, 1967) and billiards (Sinai, 1970), using Kolmogorov's idea of K-systems. Some of this was motivated by S. Smale's visit to Moscow in 1961, during which he met Anosov, Arnold and Sinai and told them of the conjecture that structurally stable systems with infinitely many periodic orbits could exist (see Smale's Horseshoe, below).

Perhaps the most influential Russian contribution to dynamical systems in this period was a "converse" to Poincaré's discovery of nonintegrability and chaos: the Kolmogorov-Arnold-Moser (KAM) Theorem. This result, which was announced at the 1954 International Congress of Mathematicians (ICM) in Amsterdam, proves the existence of a positive measure set of quasiperiodic motions lying on invariant tori for Hamiltonian flows that are sufficiently close to completely integrable systems (Kolmogorov 1957). During this period V.I. Arnold began his thesis studies with Kolmogorov. Arnold would go on to make important contributions to the quasiperiodic motion problem and in dynamical systems, bifurcation theory, and classical mechanics in general. Moser's interest in celestial mechanics had been sparked by C.L. Seigel in Göttingen and his contribution to KAM was in turn prompted by an invitation to write a summary of Kolmogorov (1957) for Mathematical Reviews. After working independently (Moser, 1962), he visited the Moscow group in 1962. This and Moser's subsequent work was instrumental in spreading the news of this important result.

While Russian mathematicians were somewhat isolated during the 1950-1980s (the height of the cold war), and were rarely able to travel outside the Soviet Bloc, they eagerly sought contacts with the West. Visits such as those of Smale and Moser, and the Moscow ICM in 1966, which Smale also attended, helped maintain communications with their European and American colleagues. Those mathematicians able to visit Moscow found a hospitable welcome.

## Radar, Nonlinear Oscillators, and Chaos

Parallel to the mathematical work of Birkhoff and others in the U.S. and Europe, interest in nonlinear oscillations was growing, driven largely by developments in electronics (indeed, Lefschetz, mentioned above, edited a series of books on the topic). B. van der Pol, a radio engineer at the Phillips labs in Eindhoven, published a remarkable short paper (van der Pol and van der Mark 1927) which may contain the first experimental observation of deterministic chaos (they describe an "irregular noise" in a diode subject to periodic forcing). This paper also noted the coexistence of periodic orbits of different period (subharmonics of two distinct orders), which implied the existence of a complicated unstable invariant set, following (Birkhoff, 1932).

Van der Pol's work was one of the motivations for Cartwright and Littlewood's (1945) study of the van der Pol equation, which in turn led to that of Levinson (1949). Levinson simplified the problem by replacing the cubic nonlinearity with a piecewise linear function, and he provided a more explicit analysis. In these papers the existence of a chaotic invariant set was proved for a periodically-forced second order ODE. It is interesting that Cartwright and Littlewood worked on the problem as part of the British program to develop radar during World War II. Meanwhile, E. Hopf, in Germany, had extended the Poincaré-Andronov bifurcation that now generally bears his name to $$n > 2$$ dimensions (Hopf 1942), prefiguring the general Center Manifold Theorem and normal form theory (Pliss 1964; Kelley 1967).

## Smale's Horseshoe

Figure 3: Steve Smale, circa 19?? (Steve, put year here).

In the late 1950's S. Smale brought a topological approach to the study of dynamical systems. Extending Andronov and Peixoto's work on structural stability to $$n > 2$$ dimensions, he defined Morse-Smale systems and conjectured that a system is structurally stable if and only if it is Morse-Smale. (A Morse-Smale system has a finite set of fixed points and periodic orbits, all of which are hyperbolic and all of whose stable and unstable manifolds intersect transversely, but no other nonwandering or recurrent points). However, after N. Levinson drew his attention to Cartwright and Littlewood's (1945) work and his own paper (Levinson, 1949), Smale abstracted a geometrical description of the chaotic set – now called the Smale horseshoe – and used it to disprove his own conjecture. He constructed an example of a structurally stable mapping on the sphere having a countable infinity of periodic orbits, and further used symbolic dynamics to describe the uncountable nonwandering Cantor set, thereby completing the story begun by Poincaré. This occurred during a visit to Peixoto's IMPA in Rio de Janeiro in 1960, close to, if not actually on the beach at Rio (Smale 1980). The paper describing it appeared five years later (Smale 1965).

Smale's work in dynamical systems became widely known only after his classic survey article (Smale 1967) (reprinted with additional notes in Smale (1980)). J. Moser (1973) subsequently gave a beautiful exposition that included explicit criteria for the presence of horseshoes in two-dimensional maps. Independently, V.K. Melnikov used regular perturbation methods to prove the existence of transverse homoclinic orbits to periodic motions in ODEs (Melnikov 1963) and V.I. Arnold generalized these ideas to produce the first example of what is now called Arnold Diffusion (Arnold 1964). In fact the (now widely-used) Melnikov function is, up to a constant, exactly the integral that Poincaré derived from Hamilton-Jacobi theory to obtain his obstruction to integrability in Poincaré (1890). Thus the circle nicely closes.

## More Chaos

The 1960's was not only a chaotic decade in mathematical and social circles; physicists and engineers were also involved. In 1959 Boris Chirikov had introduced the first analytical estimate for the onset of chaotic motion in deterministic Hamiltonian systems. Now known as the Chirikov criterion, it successfully explained puzzling experiments on plasma confinement in open magnetic traps performed at the Kurchatov Institute and later found broad applications in various physical systems. E.N. Lorenz's numerical study of a three-dimensional ODE modeling Rayleigh-Benard convection introduced sensitive dependence on initial conditions in meteorology (Lorenz 1963; see also Butterfly Effect), and around 1962 Y. Ueda demonstrated chaos in a periodically-forced oscillator using an analog computer, although his work remained unpublished for several years (Hayashi et al., 1970; Ueda 1973). Two mathematicians, David Ruelle and Floris Takens, become interested in fluid turbulence in the late 1960's and published an influential paper (Ruelle and Takens, 1971, also see Ruelle, 1991). Around the same time, J.A. Yorke discovered and publicised Lorenz's work to mathematicians, H. Swinney, J.P. Gollub and others began their experimental studies of fluid instabilities and turbulence (see, e.g. Swinney and Gollub, 1981), and a continuing festival of interdisciplinary and international conferences was launched. A notable early example was the New York Academy of Sciences meeting in November 1977 (Gurel and Rössler, 1979). The rest is more history.

This account is primarily drawn from a review article (Holmes, 2005); the slides from an illustrated companion lecture are available on DSWeb (http://tutorials.siam.org/dsweb/enoc/). A more technical account appeared in (Holmes 1990), and Chenciner (1985) reviews the field extensively in greater breadth. Barrow-Green (1997) provides a fine historical study of Poincaré's contributions. Diacu and Holmes (1996) tell an extended story, with dramatized appearances by the major actors. Aubin and Dahan Dalmedico (2002) provide a deeper sociological and cultural context, discussing the influence of the computer on both theory and experiment, and addressing contributions from physicists such as Feigenbaum (1978).

There are many textbooks on dynamical systems, spanning the range from introductory (Hirsch et al., 2004) to advanced (Guckenheimer and Holmes 1983; Wiggins 2003; Kuznetsov 2004), and from applied (Strogatz 1994) to abstract (Arnold 1983; Katok and Hasselblatt 1995). The volumes on dynamical systems in the Encyclopaedia of Mathematical Sciences (Gamkrelidze 1988-90), written by many of the major Russian contributors to the theory, are particularly valuable. An early contribution from the viewpoint of physics and Hamiltonian mechanics is Lichtenberg and Lieberman (1983).

## Acknowledgments

I am indebted to Alain Chenciner and Mark Levi for helpful suggestions, advice and references; to Ya G. Sinai for information on the Moscow School and Kolmogorov's seminar, and to an anonymous referee for the reference to Aubin and Dahan Dalmedico's paper.

## Bibliography

A.A. Andronov and L. Pontryagin. Systèmes grossiers. Dokl. Akad. Nauk. SSSR, 14, 247–251, 1937.

A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier. Theory of Bifurcations of Dynamic Systems on a Plane. Jerusalem, Israel Program for Scientific Translations,1971. (Available from the U.S. Dept. of Commerce, National Technical Information Service, Springfield, VA.)

A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier. Qualitative Theory of Second-Order Dynamic Systems. Jerusalem, Israel Program for Scientific Translations, John Wiley, New York, 1973.

A.A. Andronov, A. A. Vitt and S.E. Khaikin. Theory of Oscillators (tr. F. Immerzi from the Russian, first edition Moscow, 1937). Pergamon, London, UK, 1966.

D.V. Anosov. Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature. Proc. Steklov Inst. Math. 90, 1967.

L. Arnold. Random Dynamical Systems. Springer, Heidelberg, 1998.

V.I. Arnold. Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady, 5, 342–355, 1964.

V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York, 1983.

D. Aubin and A. Dahan Dalmedico. Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures. Historia Mathematica, 29, 273-339, 2002.

J. Barrow-Green. Poincaré and the Three Body Problem. American Mathematical Society, Providence, RI, 1997.

G.D. Birkhoff. Dynamical Systems (reprinted with an introduction by J. Moser and a preface by M. Morse, 1966). American Mathematical Society, Providence, RI, 1927.

G.D. Birkhoff. Sur quelques courbes fermees remarquables. Bull. Soc. Math. de France, 60, 1–26, 1932.

M.L. Cartwright and J.E. Littlewood. On nonlinear differential equations of the second order, I: the equation $$\ddot{y} + k(1 - y^2)\dot{y} + y = b \lambda k \cos(\lambda t + a )\ ,$$ $$k$$ large. J. London Math. Soc., 20, 180–189, 1945.

A. Chenciner. Systèmes Dynamiques différentiables. Encyclopaedia Universalis, 1985.

B.V.Chirikov, Resonance processes in magnetic traps, "At. Energ." 6, 630 (1959) (in Russian; Engl. Transl., "J. Nucl. Energy Part C: Plasma Phys." 1, 253 (1960))

B.V.Chirikov, A universal instability of many-dimensional oscillator systems, "Phys. Rep." 52, 263 (1979)

F. Diacu and P. Holmes. Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, Princeton, NJ, 1996.

M.J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys., 19, 25-52, 1978.

R.V. Gamkrelidze (Ed-in-Chief) Encyclopaedia of Mathematical Sciences: Dynamical Systems I - VII (tr. various hands from the Russian). Springer, New York, 1988-1990.

J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York, 1983.

O. Gurel and O. Rössler (Eds). Bifurcation Theory and Applications in the Scientific Disciplines. Ann. New York Acad Sci. Vol 316, New York, 1979.

C. Hayashi, Y. Ueda, N. Akamatsu and H. Itakura. On the behavior of self-oscillatory systems with external force. Trans. IECE Japan, 53A (3), 150–158, 1970.

M.W. Hirsch, S. Smale and R.L. Devaney. Differential Equations, Dynamical Systems and an Introduction to Chaos (Second edition). Academic Press/Elsevier, San Diego, CA, 2004.

P. Holmes. Poincaré, celestial mechanics, dynamical systems theory and "chaos." Physics Reports, 193 (3), 137–163, 1990.

P. Holmes. Ninety plus thirty years of nonlinear dynamics: More is different and less is more. Int. J. Bifurcation and Chaos, 15 (9), 2703-2716, 2005.

E. Hopf. Abzweigung einer periodischen Lösung von einer stationären Lösung eines differential-systems. Berichte Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl., 94, 1-22, 1942.

A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, UK, 1995.

A. Kelley. The stable, center stable, center, center unstable, and unstable manifolds. J. Diffl. Eqns., 3, 546–570, 1967.

A.N. Kolmogorov Theorie generale des sytèmes dynamiques et mecanique classique. In Proc. International Congress of Mathematicians, Amsterdam, 1954, vol. I, pp 315–333. North-Holland, Amsterdam, 1957.

Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory (third edition). Springer, New York, 2004.

S. Lefschetz. Differential Equations: Geometric Theory. Interscience Publishers, New York, 1957.

N. Levinson. A second-order differential equation with singular solutions. Annals of Mathematics, 50, 127–153, 1949.

A.J. Lichtenberg and M.A. Lieberman Regular and Chaotic dynamics. Springer, Berlin-New York, 1992

E.N. Lorenz. Deterministic non-periodic flow. J. Atmosperic Sci. 20, 130–141, 1963.

V.K. Melnikov. On the stability of the center for time-periodic perturbations. Trans. Moscow Math. Soc., 12, 1–57, 1963.

J.K. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen II. Math. Phys. Kl. 1-20, 1962.

J.K. Moser. Stable and Random Motions in Dynamical Systems. Princeton University Press, Princeton, NJ, 1973.

V.V. Nemytskii and V.V. Stepanov. Qualitative Theory of Differential Equations (tr. various hands from earlier Russian editions). Princeton University Press, Princeton, NJ, 1960.

M.M. Peixoto. Structural stability on two-dimensional manifolds. Topology, 1, 101-120, 1962.

V.A. Pliss. A reduction principle in the theory of the stability of motion. Izv. Akad. Nauk. SSSR, Mat. Ser. 28, 1297–1324, 1964.

H.J. Poincaré. Sur le problème des trois corps et les équations de la dynamique. Acta Mathematica, 13, 1–270, 1890.

H.J. Poincaré. Les méthodes nouvelles de la mécanique céleste, Vols 1-3. Gauthiers-Villars, Paris, 1892, 1893, 1899. (English translation edited by D. Goroff, published by the American Institute of Physics, New York, 1993.)

D. Ruelle. Chance and Chaos Princeton University Press, Princeton, NJ, 1991.

D. Ruelle and F. Takens. On the nature of turbulence. Commun. Math. Phys., 20, 167-192, 1971; 23, 343-344, 1971.

Ya. G. Sinai. Classical dynamical systems with countable multiplicity Lebesgue spectrum, I. Izvestia Acd. Sci. USSR, Mat. 30 (1), 15-68, 1966.

Ya. G. Sinai. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math. Surveys, 25 (2), 137-189, 1970.

S. Smale. Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, pp. 63-70, S.S. Cairns (ed). Princeton University Press, Princeton, NJ, 1965.

S. Smale. Differentiable Dynamical Systems. Bull. Amer. Math. Soc., 73, 747–817, 1967.

S. Smale. The Mathematics of Time. Springer, New York, 1980.

H.L. Swinney and J.P. Gollub (Eds). Hydrodynamic Instabilities and the Transition to Turbulence. Springer, New York, 1981 (second Edition, 1985).

Y. Ueda, N. Akamatsu and C. Hayashi. Computer simulations and non-periodic oscillations. Trans. IEICE Japan, 56A (4), 218–225, 1973.

B. van der Pol and B. van der Mark. Frequency demultiplication. Nature, 120, 363–364, 1927.

S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. (Second edition). Springer, New York, 2003.

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