Dynamical Systems (Including Chaos)

And the future is certain
Give us time to work it out

Take your favorite mathematical space. It might represent physical variables, or biological ones, or social, or simply be some abstract mathematical object, whatever those are; in general each variable will be a different coordinate in the space. Come up with a rule (function) which, given any point in the space, comes up with another point in the space. It's OK if the rule comes up with the same result for two input points, but it must deliver some result for every point (can be many-one but must be defined at every point). The combination is a discrete-time dynamical system, or a map. The space of points is the state space, the function the mapping or the evolution operator or the update rule, or any of a number of obviously synonymous phrases.

The time-evolution, the dynamics, work like this: start with your favorite point in the state space, and find the point the update rule specifies. Then go to that point --- the image of the first --- and apply the rule again. Repeat forever, to get the orbit or trajectory of the point. If you have a favorite set of points, you can follow their dynamics by applying the mapping to each point separately. If your rule is well-chosen, then the way the points in state space move around matches the way the values of measured variables change over time, each update or time-step representing, generally, a fixed amount of real time. Then the dynamical system models some piece of the world. Of course it may not model it very well, or may even completely fail in what it set out to do, but let's not dwell on such unpleasant topics, or the way some people seem not to care whether the rules they propose really model what they claim they model.

This is all for discrete-time dynamics, as I said. But real time is continuous. (Actually, it might not be. If it isn't continuous, though, the divisions are so tiny that for practical purposes it might as well be.) So it would be nice to be able to model things which change in continuous time. This is done by devising a rule which says, not what the new point in state space is, not how much all the variables change, but the rates of change of all the variables, as functions of the point in state space. This is calculus, or more specifically differential equations: the rule gives us the time-derivatives of the variables, and to find out what happens at any later time we integrate. (The rule which says what the rates of change are is the vector field --- think of it as showing the direction in which a state-space point will move.) A continuous-time dynamical system is called a flow.

In either maps or flows, there can be (and generally are) sets of points which are left unchanged by the dynamics. (More exactly, for any point in the set, there is always some point in the set which maps (or flows) into its place, so the set doesn't change. The set is its own image.) These sets are called invariant. Now, we say that a point is attracted to an invariant set if, when we follow its trajectory for long enough, it always gets closer to the set. If all points sufficiently close to the invariant set are attracted to it, then the set is an attractor. (Technically: there is some neighborhood of the invariant set whose image is contained in itself. Since the invariant set is, after all, invariant, the shrinkage has to come from non-invariant points moving closer to the invariant set.) An attractor's basin of attraction is all the points which are attracted to it.

The reasons for thinking about attractors, basins of attraction, and the like, are that, first, they control (or even are) the long-run behavior of the system, and, second, they let us think about dynamics, about change over time, geometrically, in terms of objects in (state) space, like attractors, and the vector field around attractors.

Imagine you have a one-dimensional state space, and pick any two points near each other and follow their trajectories. Calculate the percentage by which the distance between them grows or shrinks; this is the Lyapunov exponent of the system. (If the points are chosen in a technically-reasonable manner, it doesn't matter which pair you use, you get the same number for the Lyapunov exponent.) If it is negative, then nearby points move together exponentially quickly; if it is positive, they separate exponentially; if it is zero, either they don't move relative to one another, or they do so at some sub-exponential rate. If you have n dimensions, there is a spectrum of n Lyapunov exponents, which say how nearby points move together or apart along different axes (not necessarily the coordinate axes). So a multi-dimensional system can have some negative Lyapunov exponents (directions where the state space contracts), some positive ones (expanding directions) and some zero ones (directions of no or slow relative change). At least one of a flow's Lyapunov exponents is always zero. (Exercise: why?) The sum of all the Lyapunov exponents says whether the state space as a whole expands (positive sum) or contracts (negative sum) or is invariant (zero sum).

If there is a positive Lyapunov exponent, then the system has sensitive dependence on initial conditions. We can start with two points --- two initial conditions --- which are arbitrarily close, and if we wait only a very short time, they will be separated by some respectable, macroscopic distance. More exactly, suppose we want to know how close we need to make two initial conditions so that they'll stay within some threshold distance of each other for a given length of time. A positive Lyapunov exponent says that, to increase that length of time by a fixed amount, we need to reduce the initial separation by a fixed factor (the time is logarithmic in the initial separation). Now think of trying to predict the behavior of the dynamical system. We can never measure the initial condition exactly, but only to within some finite error. So the relationship between our guess about where the system is, and where it really is, is that of two nearby initial conditions, and our prediction is off by more than an acceptable amount when the two trajectories diverge by more than that amount. Call the time when this happens the prediction horizon. Sensitive dependence says that adding a fixed amount of time to the prediction horizon means reducing the initial measurement error by a fixed factor, which quickly becomes hopeless. More optimistically, if we re-measure where the system is after some amount of time, we can work back to say more exactly where the initial condition was. To reduce the (retrospective) uncertainty about the initial condition by a fixed factor, wait a fixed amount of time before re-measuring...

Sensitive dependence is not, by itself, dynamically interesting; very trivial, linear dynamical systems have it. (Exponential growth, for instance!) Something like it has been appreciated from very early times in dynamics. Laplace, for instance, so often held up to ridicule or insult as a believer in determinism and predictability fully recognized that (to use the modern jargon) very small differences in initial conditions can have very large effects, and that our predictions are correspondingly inexact and uncertain. That's why he wrote books on probability theory! And as a proverb, the butterfly effect ("The way a butterfly flaps its wings over X today can change whether or not there's a hurricane over Y in a month") isn't really much of an improvement over "For want of a nail, a horse was lost". (It did, however, inspire Terry Pratchett's fine comic invention, the Quantum Chaos Butterfly, which causes small hurricanes to appear when it flaps its wings.) No, what's dynamically interesting is the combination of sensitive dependence and some kind of limit on exponential spreading. This could be because the state space as a whole is bounded, or because the sum of the Lyapunov exponents is negative or zero. That, roughly speaking, is chaos. (There are much more precise definitions!) In particular, if the sum of the Lyapunov exponents is negative, but some are positive, then there is an attractor, with exponential separation of points on the attractor --- called, for historical reasons, a strange attractor.

Chaotic systems have many fascinating properties, and there is a good deal of evidence that much of nature is chaotic; the solar system, for instance. (This is actually, by a long and devious story, where dynamical systems theory comes from.) It raises a lot of neat and nasty problems about how to understand dynamics from observations, and about what it means to make a good mathematical model of something. But it's not the whole of dynamics, and in some ways not even the most interesting part, and it's certainly not the end of "linear western rationalism" or anything like that.

Things I ought to talk about here: Time series. Geometry from a time series/attractor reconstruction. (History. The dominant citation is to Takens's 1981 paper, which proved that it works generically. However, Takens disclaims introducing the method. The Santa Cruz group --- Packard, Crutchfield, Farmer and Shaw --- had the first publication, in 1980. Since Crutchfield was my adviser, I'd like to think they came up with it. But they attribute the idea of using time-lags to personal communication from Ruelle (note 8 in their paper), who seems to be the actual originator.) Symbolic dynamics. Structural stability. Bifurcations. The connection to fractals. Spatiotemporal chaos.

Uses and abuses: Military uses. Popular and semi-popular views. Metaphorical uses. Appropriation by non-scientists.

Stephen Kellert
- In the Wake of Chaos [Discusses the (modest) philosophical import of chaos. Great opening: "Chaos theory is not as interesting as it sounds. How could it be?"]
- "Science and Literature and Philosophy: The Case of Chaos Theory and Deconstruction", Configurations 1996 2:215 [tho' he's not nearly as harsh on Hayles or Arygros as they deserve, and he really ought to read Gross and Levitt more carefully]
Pierre-Simon Laplace, Philosophical Essay on Probabilities, Part I
Henri Poincaré Science and Method, ch. 4, "Chance" [Soon, with a bit of luck, to be on-line]
David Ruelle, Chance and Chaos [An account of chaos from one of those "present at the creation"; a jewel]
Ian Stewart, Does God Play Dice? [Probably the best popular book, certainly the one which tells you the most about what the field is actually about.]

Abraham and Shaw, Dynamics: The Geometry of Behavior [An entirely visual approach to teaching dynamics; all the equations live in a ghetto-appendix, if you really want to see them. Abraham, sad to say, seems to have flipped his lid, and published a book called Chaos Gaia Eros, tracing chaos theory back through "25,000 years of Orphic tradition" on the basis of cranks of the sort satirized by Umberto Eco, to say nothing of revelations in the Himalayas. Remember, children, drugs are your friends: always treat them with respect, and they make life better; abuse them, and they will let you make an ass of yourself in public.]
Baker and Gollub, Chaotic Dynamics
Pierre Berge et al., Order within Chaos
Robert Devaney
- A First Course in Chaotic Dynamical Systems [Less advanced]
- Introduction to Chaotic Dynamical Systems [More advanced]
Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation [Review: A Garden of Bright Images]
Andrew M. Fraser, Hidden Markov Models and Dynamical Systems
David Ruelle, "Determinstic Chaos: The Science and the Fiction", Proceedings of the Royal Society of London A 427 (1990): 241--248 [JSTOR]
Peter Smith, Explaining Chaos [Nice presentation of the basics of chaos, plus discussion of why their philosophical import is even smaller than Kellert allows]
Thomas Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem. [Detailed technical history of the interaction of analytical math and simulation in the FPU problem, the first important problem in dynamics to be attacked by simulation; and fairly unhelpful and obvious philosophical ruminations on the methodological role and status of simulation]
Charlotte Werndl, "Deterministic versus indeterministic descriptions: not that different after all?", pp. 63--78 in A. Hieke and H. Leitgeb (eds.), Reduction, Abstraction, Analysis: Proceedings of the 31st International Ludwig Wittgenstein-Symposium = phil-sci/4775

On-line archives:
- nlin.CD, formerly chao-dyn, for chaotic dynamics
- math.DS, for dynamical systems
H. D. I. Abrabanel, Analysis of Observed Chaotic Data
V. I. Arnol'd
- Catastrophe Theory [Warning: this book is very light on equations, but very heavy on the mathematical knowledge it demands.]
- Mathematical Methods of Classical Mechanics
- Ordinary Differential Equations [Introductory book on ODEs which presents them the right way, as dynamical systems.]
June Barrow-Green, Poincaré and the Three Body Problem [Historical]
Beck and Schlögl, Themodynamics of Chaotic Systems. [Formal analogies between chaos and statistical mechanics, which give you ways of calculating dimensions, Lyapunov exponents, entropies, etc., and showing connections between them. (There's no known link between chaos in general and physical thermodynamics.) I got my copy when visiting my brother at his summer internship in Pittsburgh in '95. We'd gone to the science museum (which like everything else is in the city is named after Carnegie) to see an Imax movie about sharks, and play hob with my inner ears. In the giftshop, cheek-by-jowl with pocket guides to astronomy and one of Gonick's Cartoon Guides, was this book, which is vol. 4 in Cambridge's Nonlinear Science Series, and a graduate-level physics text which assumes at least some familiarity with thermodynamics, statistical mechanics, fractals, chaotic dynamics and measure theory. Now it's a very good textbook, but I think we have to conclude that either (i) the inhabitants of Pittsburgh are so well-educated it's not even funny or (ii) this whole chaotophilia business has gone altogether too far. N.B. the museum was not also selling, say, Griffiths' Introduction to Electrodynamics.]
P.-M. Binder and Milena C. Cuéllar, "Chaos and Experimental Resolution," Physical Review E 61 (2000): 3685--3688
G. Boffetta, M. Cencini, M. Falcioni and A. Vulpiani, "Predictability: a way to characterize Complexity," nlin.CD/0101029
M. Cencini, M. Falconi, Holger Kantz, E. Olbrich and Angelo Vulpiani, "Chaos or Noise: Difficulties of a Distinction," Physical Review E 62 (2000): 427--437 = nlin.CD/0002018
J.-R. Chazottes and F. Redig, "Testing the irreversibility of a Gibbsian process via hitting and return times", math-ph/0503071
J.-R. Chazottes and E. Uglade, "Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources", math.DS/0401093
F. K. Diakonos, D. Pingel and P. Schmelcher, "A Stochastic Approach to the Construction of One-Dimensional Chaotic Maps with Prescribed Statistical Properties," chao-dyn/9910020
J. R. Dorfman, Introduction to Chaos in Nonequilibrium Statistical Mechanics [A dual to Beck and Schlögl --- how chaos is useful in giving us statistical mechanics. New and elegant approaches to the old problem of why it should be valid to treat a large, deterministic mechanical system statistically.]
Freidlin and Wentzell, Random Perturbations of Dynamical Systems [See under large deviations]
Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
Holger Kantz and Thomas Schreiber, Nonlinear Time Series Analysis
Vivien Lecomte, Cecile Appert-Rolland and Frederic van Wijland, "Chaotic properties of systems with Markov dynamics", cond-mat/0505483 = Physical Review Letters 95 (2005): 010601 [Showing that the thermodynamic formalism can work for continuous-time Markov processes, which is very nice]
James Ramsay, Giles Hooker, David Campbell and Jiguo Cao, "Parameter Estimation for Differential Equations: A Generalized Smoothing Approach", Journal of the Royal Statistical Society forthcoming (2007) [PDF preprint]
David Ruelle, Chaotic Evolution and Time-Series
O. Shenker, "Fractal geometry is not the geometry of nature," Studies in the History and Philosophy of Science 25 (1994): 967--981
Benjamin Weiss, Single Orbit Dynamics

James Gleick, Chaos: The Making of a New Science [Yes, I'm completely serious about dis-recommending this. Get hold of Ian Stewart's book above, instead.]
N. Katherine Hayles. Prof. Hayles is the head of the Science and Literature section of the Modern Language Association. She writes books (e.g., Chaos Bound) alleging a profound connection between chaos, complexity, etc., and deconstruction and other strains of the French Disease. I cannot (professionally) speak for her understanding of Derrida et cie, but if she has any understanding of the science, she has kept it well-hidden. [Argyros below is supposed to be better. I have my doubts.]

Argyros, A Blessed Rage for Order: Deconstruction, Evolution, and Chaos
Baker, Centring the Periphery: Chaos, Order, and the Ethnohistory of Dominica [McGill University Press. These people may be kooks, but they're not crackpots.]
Depew and Weber, Darwinism Evolving [Review by John Maynard Smith]
Florin Diacu and Philip Holmes, Celestial Encounters: The Origins of Chaos and Stability
Eugene Eoyang, "Chaos Misread", Comparative Literature Studies, 1989 26:271
Angus Fletcher, A New Theory of American Poetry: Democracy, the Environment, and the Future of Imagination [Recommended, in this connection, by a correspondent who prefers to remain nameless]
Freund, Broken Symmetries: a Study of Agency in Shakespeare's Plays [Am I alone in thinking that this book --- which is listed under "Chaotic behavior in systems" in the library catalog --- is going to prove to be really horrible?]
Gordon E. Slethaug, Beautiful Chaos: Chaos Theory and Metachaotics in Recent American Fiction
Zel'dovich et al., Almighty Chance

M. Abel, L. Biferale, M. Cencini, M. Falconi, D. Vergni and A. Vulpiani
- "An Exit-Time Approach to \epsilon-Entropy," chao-dyn/9912007
- "Exit-Times and \epsilon-Entropy for Dynamical Systems, Stochastic Processes, and Turbulence," nlin.CD/0003043
D. J. Albers, J. C. Sprott and J. P. Crutchfield, "Persistent Chaos in High Dimensions", nlin.CD/0504040
P. Allegrini, V. Benci, P. Grigolini, P. Hamilton, M. Ignaccolo, G. Menconi, L. Palatella, G. Raffaelli, N. Scafetta, M. Virgilio and J. Jang, "Compression and diffusion: a joint approach to detect complexity," cond-mat/0202123
Vitor Araujo, "Random Dynamical Systems", math.DS/0608162 = pp. 330--385 in J.-P. Francoise, G. L. Naber and Tsou S. T. (eds.), Encyclopedia of Mathematical Physics, vol. 3
Arnol'd and Avez, Ergodic Problems of Classical Mechanics
Bidhan Chandra Bag, Jyotipratim Ray Chaudhuri and Deb Shankar Ray, "Chaos and Information Entropy Production," chao-dyn/9908020
Baptista, Rosa and Greborgi, "Communication through Chaotic Modeling of Languages," Physical Review E 61 (2000): 3590--3600
V. I. Bakhtin, "Positive Processes", math.DS/0505446 ["we introduce positive flows and processes, which generalize the ordinary dynamical systems and stochastic processes", with promises of laws of large numbers, large deviation properties and action functionals]
Barnsley, Fractals Everywhere 2nd ed. [Yes, yes, I know, it's not really chaos.]
Jacopo Bellazzini, "Holder regularity and chaotic attractors," nlin.CD/0104013
Nils Berglund
- "Geometrical theory of dynamical systems," math.HO/0111177
- "Perturbation theory of dynamical systems," math.HO/0111178
George D. Birkhoff, Dynamical Systems [1927; online]
Claudio Bonanno, "The Manneville map: topological, metric and algorithmic entropy," math.DS/0107195
Joseph L. Breeden and Alfred Hübler, "Reconstructing Equations of Motion from Experimental Data with Unobserved Variables," Physical Review E 42 (1990): 5817--5826
Benoit Cadre and Pierre Jacob, "On Symmetric Sensitivity", math.DS/0501222
P. Castiglione, M. Falcioni, A. Lesne and A. Vulpiani, Chaos and Coarse Graining in Statistical Mechanics [Blurb, Review in J. Stat. Phys.]
Jean-René Chazottes and Bastien Fernandez (eds.), Dynamics of Coupled Map Lattices and Related Spatially Extended Systems [Blurb; 9Mb PDF preprint]
Piero Cipriani and Antonio Politi, "An open-system approach for the characterization of spatio-temporal chaos," nlin.CD/0301003
Nguyen Dinh Cong, Topological Dynamics of Random Dynamical Systems
Pedrag Cvitanovic
- "Chaotic Field Theory: A Sketch," nlin.CD/0001034
- (ed.) Universality in Chaos
Shmuel Fishman and Saar Rahav, "Relaxation and Noise in Chaotic Systems," nlin.CD/0204068
Sara Franceschelli [History of experimental application of nonlinear dynamics ideas; thesis on the development and implementation of the idea of intermittency. All publications may be in French, though]
Roman Frigg, "In What Sense is the Kolmogorov-Sinai Entropy a Measure for Chaotic Behaviour? - Bridging the Gap Between Dynamical Systems Theory and Communication Theory", phil-sci/2929 = British Journal for the Philosophy of Science 55 (2004): 411--434 [It seems to me to be obvious by definition that the Kolmogorov-Sinai entropy is a (supremum over) Shannon entropy rates, so presumably there is more going on here than is shown by the abstract]
Gary Froyland, "Statistical optimal almost-invariant sets", Physica D 200 (2005): 205--219 [Partitioning state space into nearly separated components.]
Stefano Galatolo, "Information, initial condition sensitivity and dimension in weakly chaotic dynamical systems," math.DS/0108209
F. Ginelli, R. Livi and A. Politi, "Emergence of chaotic behaviour in linearly stable systems," nlin.CD/0102005
F. Ginelli, P. Poggi, A. Turchi, H. Chate, R. Livi, and A. Politi, "Characterizing Dynamics with Covariant Lyapunov Vectors", Physical Review Letters 99 (2007): 130601
Glass and Mackey, From Clocks to Chaos
Sebastian Gouzel, "Decay of correlations for nonuniformly expanding systems", math.DS/0401184
Tilmann Gneiting and Martin Schlather, "Stochastic Models Which Separate Fractal Dimension and Hurst Effect," physics/0109031
Gilad Goren, Jean-Pierre Eckmann, and Itamar Procaccia, "Scenario for the Onset of Space-Time Chaos," Physical Review E 57 (1998): 4106--4134
A. Greven, G. Keller and G. Warnecke (eds.), Entropy
Emilio Hernandez-Garcia, Cristina Masoller, and Claudio R. Mirasso, "Anticipating the Dynamics of Chaotic Maps," nlin.CD/0111014
Kevin Judd, "Failure of maximum likelihood methods for chaotic dynamical systems", Physical Review E 75 (2007): 036210 [He means failure for state estimation, not parameter estimation. I wonder if this isn't linked to the old Fox & Keizer papers about amplifying fluctuations in macroscopic chaos?]
Kunihiko Kaneko and Ichiro Tsuda, "Chaotic Itinerancy", Chaos 13:3 (2003): 926--936 [Introduction to a special issue on the subject. "Chaotic itinerancy is ... itinerant motion among varieties of low-dimensional ordered states through high-dimensional chaos."]
Holger Kantz and Thomas Schuermann, "Enlarged scaling ranges for the KS-entropy and the information dimension," Chaos 6 (1996): 167--171 = cond-mat/0203439
Hans G. Kaper and Tasso J. Kaper, "Asymptotic Analysis of Two Reduction Methods for Systems of Chemical Reactions," math.DS/0110159 [Reduction in the mathematical, not the chemical, sense!]
Katok and Hasselblatt, Modern Dynamical Systems Theory
S. Kriso, R. Friedrich, J. Peinke and P. Wagner, "Reconstruction of dynamical equations for traffic flow," physics/0110084
Vito Latora and Michel Baranger, "Kolmogorov-Sinai Entropy-Rate vs. Physical Entropy," chao-dyn/9806006
Stefano Luzzatto, "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results," math.DS/0301319
Anil Maybhate, R. E. Amritkar and D. R. Kulkarni, "Estimation of Initial Conditions and Secure Communication," nlin.CD/011003
Sonnet Q. H. Nguyen and Lukasz A. Turski, "On the Dirac Approach to Constrained Dissipative Dynamics," physics/0110065
D. S. Ornstein and B. Weiss, "Statistical Properties of Chaotic Systems," Bulletin of the American Mathematical Society 24 (1991): 11--116
Guillermo Ortega, Cristian Degli Esposti Boschi and Enrique Louis, "Detecting Determinism in High Dimensional Chaotic Systems," nlin.CD/0109017
P. Palaniyandi and M. Lakshmanan, "Estimation of System Parameters and Predicting the Flow Function from Time Series of Continuous Dynamical Systems", nlin.CD/0406027
Nita Parekh and Somdatta Sinha, "Controlling Spatiotemporal Dynamics in Excitable Systems," SFI Working Paper 00-06-031
Luc Pronzato et al., Dynamical Search: Applications of Dynamical Systems in Search and Optimization
Ramiro Rico-Martinez, K. Krischer, G. Flaetgen, J.S. Anderson and I.G. Kevrekidis, "Adaptive Detection of Instabilities: An Experimental Feasibility Study," nlin.CD/0202057
James C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors
Jacek Serafin, "Finitary Codes, a short survey", math.DS/0608252
Eduardo D. Sontag, "For differential equations with r parameters, 2r+1 experiments are enough for identification," math.DS/0111135
Strogatz, Nonlinear Dynamics and Chaos [Good undergraduate textbook for applications; not finished with it yet]
Kazumasa A. Takeuchi, Francesco Ginelli and Hugues Chaté, "Lyapunov Analysis Captures the Collective Dynamics of Large Chaotic Systems", Physical Review Letters 103 (2009): 154103 = arxiv:0907.4298
Julien Tailleur and Jorge Kurchan, "Probing rare physical trajectories with Lyapunov weighted dynamics", cond-mat/0611672 ["we implement an efficient method that allows one to work in higher dimensions by selecting trajectories with unusual chaoticity"]
Naoki Tanaka, Hiroshi Okamoto and Masayoshi Naito, "Estimating the active dimension of the dynamics in a time series based on an information criterion," Physica D 158 (2001): 19--31
Sorin Tanase-Nicola and Jorge Kurchan, "Statistical-mechanical formulation of Lyapunov exponents," cond-mat/0210380
Ioana Triandaf, Erik M. Bollt and Ira B. Schwartz, "Approximating stable and unstable manifolds in experiments," Physical Review E 67 (2003): 037201
H. White, "Algorithmic Complexity of Points in a Dynamical System", Ergodic Theory and Dynamical Systems 13 (1993): 807
Damian H. Zanette and Alexander S. Mikhailov, "Dynamical systems with time-dependent coupling: clustering and critical behavior", Physica D 194 (2004): 203--218

CRS, "Complexity and Entropy on Routes to Chaos"

Notebooks

Dynamical Systems (Including Chaos)