## Dynamical Systems (Including Chaos)

*13 Dec 2015 13:39*

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.

See also: Cellular Automata; Complexity; Complexity Measures; Computational Mechanics; Ergodic Theory; Evolution; Information Theory [the sum of the positive Lyapunov exponents is the rate of information production]; Machine Learning, Statistical Inference and Induction; Math I Ought to Learn; Neuroscience; Pattern Formation; Philosophy of Science; Probability; Self-Organization; Simulation; Statistics; Statistical Mechanics; Synchronization; Time Series, or Statistics for Stochastic Processes and Dynamical Systems; Turbulence

- Recommended, non-technical:
- 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.]

- Recommended, technical but introductory-level:
- 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
- M. S. Bartlett, "Chance or Chaos?", Journal of the Royal
Statistical Society A
**153**(1990): 321--347 [JSTOR] - 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

- Recommended, technical and advanced:
- On-line archives:
- D. J. Albers, Fatihcan M. Atay, "Entropy, dimension, and state mixing in a class of time-delayed dynamical systems", arxiv:0710.2626
- D. J. Albers, J. C. Sprott and J. P. Crutchfield, "Persistent Chaos in High Dimensions", nlin.CD/0504040
- 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

- Dis-recommended:
- 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.]

- To read, popularization, history, philosophy, and appropriations:
- 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?]
- Laurent Mazliak, "Poincarés Odds", arxiv:1211.5737
- Gordon E. Slethaug, Beautiful Chaos: Chaos Theory and Metachaotics in Recent American Fiction
- Zel'dovich
*et al.,*Almighty Chance

- To read, technical:
- 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

- 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
- Roberto Artuso, Cesar Manchein, "Instability statistics and mixing rates", arxiv:0906.0791
- 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]
- M. S. Baptista, R. M. Rubinger, E. R. V. Junior, J. C. Sartorelli, U. Parlitz, C. Grebogi, "Upper and lower bounds for the mutual information in dynamical networks", arxiv:1104.3498
- 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
- Claudio Bonnano and Pierre Collet, "Complexity for Extended
Dynamical Systems", Communications
in Mathematical Physics
**275**(2007): 721--748 - 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

- Tomasz Downarowicz, Entropy in Dynamical Systems
- David P. Feldman, Chaos and Fractals: An Elementary Introduction [Blurb. Dave is an old friend who taught me much when we were both graduate students at SFI.]
- 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 that not only is it pretty much*obvious by definition*that the Kolmogorov-Sinai entropy is a (supremum over) Shannon entropy rates, but that various textbooks (e.g., Keane's or Sinai's) prove the supremum is actually attained for generating partitions, 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
- Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas, "Dynamical systems, simulation, abstract computation", arxiv:1101.0833
- 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 - Leon Glass and Michael C. Mackey, From Clocks to Chaos
- Tilmann Gneiting, Hana Ševčíková, and Donald B. Percival, "Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data", Statistical Science
**27**(2012): 247--277 - 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
- Tilmann Gneiting, Hana Sevcikova, Donald B. Percival, "Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data", arxiv:1101.1444
- 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
- Steven Huntsman, "Effective statistical physics of Anosov systems", arxiv:1009.2127
- 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
- Clemens Kreutz, Andreas Raue, Jens Timmer, "Likelihood based observability analysis and confidence intervals for predictions of dynamic models", arxiv:1107.0013
- 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
- Y. Charles Li, "Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence", arxiv:0712.4026
- Stefano Luzzatto, "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results," math.DS/0301319
- Cesar Maldonado, "Fluctuation Bounds for Chaos Plus Noise in Dynamical Systems", Journal of Statistical Physics
**148**(2012): 548--564 - Anil Maybhate, R. E. Amritkar and D. R. Kulkarni, "Estimation of Initial Conditions and Secure Communication," nlin.CD/011003
- Kevin McGoff, Sayan Mukherjee, Natesh S. Pillai, "Statistical inference for dynamical systems: a review", arxiv:1204.6265
- 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 - Divakar Viswanath, Xuan Liang, Kirill Serkh, "Metric Entropy and the Optimal Prediction of Chaotic Signals", arxiv:1102.3202
- H. White, "Algorithmic Complexity of Points in a Dynamical
System", Ergodic Theory and Dynamical Systems
**13**(1993): 807 - B. D. Wissman, L. C. McKay-Jones, and P.-M. Binder, "Entropy rate estimates from mutual information", Physical Review E
**84**(2011): 046204 - Damian H. Zanette and Alexander S. Mikhailov, "Dynamical systems
with time-dependent coupling: clustering and critical behavior", Physica
D
**194**(2004): 203--218

- To write:
- CRS, "Complexity and Entropy on Routes to Chaos"