Symbolic Dynamics

A useful method for studying discrete-time dynamical systems with continuous state spaces. The basic idea is to take the state space and partition it into a finite number of regions, each of which you label with some symbol. Each point in the state space then gives you an infinite sequence of symbols: the symbol for the cell of the partition of the original point, the symbol for the cell of its first iterate, its second, and so forth. Normally this involves some loss of information, since you're going from continuous to discrete values. The symbolic dynamics are often stochastic even when the continuous dynamics are deterministic, for starters. So why do this?

One reason is that the dynamics as such are drastically simplified. Let me introduce some notation here. Write for the continuous state, for the map, and for the function taking states to symbols. Then the symbol sequence . Now suppose I started with instead; my sequence would be . So the dynamics on the space of symbol sequences just consists of shifting to the right: . (This is called the shift map.) This is completely trivial. All of the structure in the problem has been shifted from the map to space of sequences. For instance, certain subsequences may not occur at all, or differ in probability. But we have a lot of tools for studying sets of discrete symbol sequences --- not just stochastic process theory, but the theory of formal languages, for instance, so that we can write out the abstract automata which generate the symbol sequences produced by the dynamics.

The other reason for using symbolic dynamics is that there are important cases where one can find generating partitions --- mappings into symbols where there is a one-to-one correspondence between continuous states and the symbol sequences they generate. In these cases, studying the symbolic dynamics is completely equivalent to studying the original dynamics. Remarkably enough, even with a generating partition, the symbolic dynamics can be stochastic for an underlying deterministic system. In this way, for instance, one can show that some kinds of (sufficiently chaotic) deterministic dynamics are in a sense completely equivalent to sources of independent, identically-distributed random variables.

To go off on a tangent, there's something of a movement in cognitive science which sets up an opposition between computation, conceived in the usual symbol-manipulation sense, and continuous nonlinear dynamics. This seems to me quite wrong-headed, if only because the existence of generating partitions shows how symbol-manipulating computation can be completely equivalent to a dynamical system. Computation is intrinsic to dynamics, but that's another topic.

The most fundamental reason I like symbolic dynamics is that I know a lot of tricks for predicting discrete stochastic processes, and want to exploit them as widely as possible.

Cf. Computation, Automata and Formal Languages; Computational Mechanics; Dynamics in Cognitive Science; Ergodic Theory; Information Theory; Nonlinear Dynamics; Time Series

R. L. Adler and B. Weiss, "Entropy, a Complete Metric Invariant for Automorphisms of the Torus", Proceedings of the National Academy of Sciences (USA) 57 (1967): 1573--1576 [PDf reprint. Classic and cute, if somewhat specialized, paper.]
Remo Badii and Antonio Politi, Complexity: Hierarchical Structures and Scaling in Physics [Review, with some more explanation]
C. Beck and F. Schögl, Thermodynamics of Chaotic Systems [Applying the thermodynamic formalism to symbolic dynamics]
Peter beim Graben and Harald Atmanspacher, "Complementarity in Classical Dynamical Systems", nlin.CD/0407046 [A really cool, if rather counter-intuitive, result about how one can get something which looks rather like quantum-mechanical "complementarity" (i.e., incompatible observables) from differing symbolic partitions of a single classical dynamical system. I hasten to add that this is not how I think quantum mechanics works, and I'm pretty sure it's not how beim Graben and Atmanspacher think it works.]
P.-M. Binder and Juan M. Pedraza, "Nonregular Languages in the Kicked Rotor," Physical Review E 62 (2000): R5883--R5886
Erik M. Bollt, Theodore Stanford, Ying-Cheng Lai and Karol Zyczkowski, "What Symbolic Dynamics Do We Get with a Misplaced Partition? On the Validity of Threshold Crossings Analysis of Chaotic Time-Series," Physica D 154 (2001): 259--286 [Short answer: You get really wrong symbolic dynamics.]
Milena C. Cuellar and P.-M. Binder, "Reducing Noise in Discretized Time Series", Physical Review E 64 (2001): 046211
Ruslan L. Davidchack, Ying-Cheng Lai, Erik M. Bollt and Mukeshwar Dhamala, "Estimating Generating Partitions of Chaotic Systems by Unstable Periodic Orbits," Physical Review E 61 (2000): 1353--1356
R. L. Devaney, A First Course in Chaotic Dynamical Systems
Stefano Galatolo, Mathieu Hoyrup, and Cristóbal Rojas, "Effective symbolic dynamics, random points, statistical behavior, complexity and entropy", arxiv:0801.0209 [All, not almost all, Martin-Lof points are statistically typical.]
Yoshito Hirata, Kevin Judd and Kazuyuki Aihara, "Characterizing chaotic response of a squid axon through generating partitions", Physics Letters A 346 (2005): 141--147
Yoshiro Hirata, Kevin Judd and Devin Kilaminster, "Estimating a generating partition from observed time series: Symbolic shadowing", Physical Review E 70 (2004): 016215 [A very elegant answer to "how do we actually find a generating partition then?"]
Matthew B. Kennel and Michael Buhl, "Estimating good discrete partitions from observed data: symbolic false nearest neighbors", nlin.CD/0304054 = Physical Review Letters 91 (2003): 084102
Bruce P. Kitchens, Symbolic Dynamics [More algebraic and less dynamical or probabilistic than I'd hoped. But useful.]
Douglas Lind and Brian Marcus, Introduction to Symbolic Dynamics and Coding

Elena S. Dimitrova, John J. McGee, and Reinhard C. Laubenbacher, "Discretization of Time Series Data", q-bio.OT/0505028 [There are some interesting ideas here, and I like that they tested the ability of their discretization method to preserve information (in some sense) within and across time-series. But they don't compare its ability to preserve information with other discretization schemes (say, applying randomly-chosen cut-offs), and gave me no sense of why the scheme itself should work.]

Jon Aaronson and Hotoshi Nakada, "Exchangeable, Gibbs and equilibrium measures for Markov subshifts", math.PR/0505011
Fatihcan Atay, Sarika Jalan and Jürgen Jost, "Randomness, chaos, and structure", arxiv:0711.4293
Rajeev K. Azad, J. Subba Rao, and Ramakrishna Ramaswamy, "Symbol sequence analysis of climatic time signals", Nonlinear Analysis: Real-World Applications 5 (2004): 487--500
Peter beim Graben, J. Douglas Saddy, Matthias Schlesewsky and Jürgen Kurths, "Symbolic Dynamics of Event-Related Brain Potentials," Physical Review E 62 (2000): 5518--5541
Camillo Cammarota and Enrico Rogora, "Spectral and symbolic analysis of heart rate data during the tilt test", Physical Review E 74 (2006):
Julien Cervelle, Enrico Formenti, Pierre Guillon, "Sofic Trace of a Cellular Automaton", math.DS/0703241 [When can the sequence of states at a particular site in a 1D cellular automaton give us a sofic shift, in the sense?]
J.-R. Chazottes, L. Ramirez and E. Ugalde, "Finite type approximations of Gibbs measures on sofic subshifts", Nonlinearity 18 (2005): 445--463 [PDF reprint via Dr. Ugalde]
Alex Clark and Lorenzo Sadun, "When size matters: subshifts and their related tiling spaces," math.DS/0201152
Ethan M. Coven and Zbigniew H. Nitecki, "On the genesis of symbolic dynamics as we know it", math.DS/0611322
Jean-Charles Delvenne, Petr Kurka and Vincent Blondel, "Computational Universality in Symbolic Dynamical Systems", cs.CC/0404021
J. M. Finn, J. D. Goettee, Z. Toroczkai, M. Anghel and B. P. Wood, "Estimation of entropies and dimensions by nonlinear symbolic time series analysis", Chaos 13 (2003): 444--456
Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas, "Effective symbolic dynamics, random points, statistical behavior, complexity and entropy", arxiv:0801.0209 ["consider the dynamical behavior of Martin-Lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. ... such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sinai entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy."]
Bai-lin Hao
- Applied Symbolic Dynamics
- "Applied Symbolic Dynamics," chao-dyn/9806025 [Presumably a resume of the book]
Yoshito Hirata and Kevin Judd, "Constructing dynamical systems with specified symbolic dynamics", Chaos 15 (2005): 033102 [explains "how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms."]
Sarika Jalan, Fatihcan M. Atay and Jürgen Jost, "Detection of synchronised chaos in coupled map networks using symbolic dynamics", nlin.CD/0510057
Anders Johansson, Anders Oberg, and Mark Pollicott, "Countable state shifts and uniqueness of g-measures", math.DS/0509109
Svetlana Katok and Ilie Ugarovici, "Symbolic Dynamics for the Modular Surface and Beyond", Bulletion of the American Mathematical Scoeity 44 (n.s., 2007): 87--132 [PDF]
Ljupco Kocarev and David M. Walker, "Compactness of Symbolic Sequences from Chaotic Systems," Physics Letters A 274 (2000): 200--205
Christophe Letellier, "Estimating the Shannon Entropy: Recurrence Plots versus Symbolic Dynamics", Physical Review Letters 96 (2006): 254102
Diana A. Mendes, Vivaldo M. Mendes, J. Sousa Ramos, "Symbolic Dynamics in a Matching Labour Market Model",nlin.CD/0608002
George Osipenko, Lectures on Symbolic Analysis of Dynamical Systems [Online PDF]
Shou-Li Peng and Ke-Fei Cao, Star Products in One-Dimensional Symbolic Dynamics
Marcus Pivato, "Cellular Automata vs. Quasistrumian Shifts", Ergodic Theory and Dynamical Systems forthcoming (2005) = math.DS/0503502
E. Arthur Robinson and Ayse A. Sahin, "On the Existence of Markov Partitions for Actions", Journal of the London Mathematical Society 69 (2004): 693--706
Ben-Zion Rubshtein, "On a class of one-sided Markov shifts", math.DS/0406059
Peter I. Saparin, Wolfgang Gowin, Jürgen Kurths, and Dieter Felsenber, "Quantification of cancellous bone structure using symbolic dynamics and measures of complexity", Physical Review E 58 (1998): 6449--6459 [Journal link]
Roy Wilds and Leon Glass, "Contrasting methods for symbolic analysis of biological regulatory networks", Physical Review E 80 (2009): 062902
H.-M. Xie, Grammatical Complexity and One-Dimensional Dynamical Systems
Liqiang Zhu, Ying-Cheng Lai, Frank C. Hoppensteadt and Erik M. Bollt, "Numerical and experimental investigation of the effect of filtering on chaotic symbolic dynamics", Chaos 13 (2003): 410--419

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