"Since the 1990s chaotic cat maps are widely used in data encryption, for their very complicated dynamics within a simple model and desired characteristics related to requirements of cryptography. The number of cat map parameters and the map period length after discretization are two major concerns in many applications for security reasons. In this paper, we propose a new family of 36 distinctive 3D cat maps with different spatial configurations taking existing 3D cat maps [1]-[4] as special cases. Our analysis and comparisons show that this new 3D cat maps family has more independent map parameters and much longer averaged period lengths than existing 3D cat maps. The presented cat map family can be extended to higher dimensional cases."

(to_teach tags for clsses which use the cat map as an example)

"The topic of statistical inference for dynamical systems has been studied extensively across several fields. In this survey we focus on the problem of parameter estimation for non-linear dynamical systems. Our objective is to place results across distinct disciplines in a common setting and highlight opportunities for further research."

"Influence systems form a large class of multiagent systems designed to model how influence, broadly defined, spreads across a dynamic network. We build a general analytical framework which we then use to prove that, while sometimes chaotic, influence dynamics is almost always asymptotically periodic. Besides resolving the dynamics of a popular family of multiagent systems, the other contribution of this work is to introduce a new type of renormalization-based bifurcation analysis for multiagent systems."

"We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. The purpose of the present paper is to show that the homogenization method developed in Imkeller et al. (2011), which is applicable to high dimensional multi-scale filtering problems, along with important sampling and control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes."

"The existence of invariant Borel probability measures for random dynamical systems on complete metric spaces is proved under assumptions that the systems have countably many maps and admit finite Markov partitions such that the resulting Markov systems are uniformly continuous and contractive, and satisfy some integrability condition in the infinite case. A one-to-one map between these measures and equilibrium states associated with such systems is established. Some properties of the map and the measures are given."

"Stochastic partial differential equations driven by Poisson random measures (PRM) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equations (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRM, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway."

"Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A."

Sweet. (Bayesian estimation seems like overkill here however, especially since predictions are just made from point estimates.)

"We study weak convergence of empirical processes of dependent data, indexed by classes of functions. We obtain results that are especially suitable for data arising from dynamical systems and Markov chains, where the Central Limit Theorem for partial sums is commonly derived via the spectral gap technique. Our results apply, e.g. to the empirical process of ergodic torus automorphisms."

"Echo state networks (ESNs) are large, random recurrent neural networks with a single trained linear readout layer. Despite the untrained nature of the recurrent weights, they are capable of performing universal computations on temporal input data, which makes them interesting for both theoretical research and practical applications. The key to their success lies in the fact that the network computes a broad set of nonlinear, spatiotemporal mappings of the input data, on which linear regression or classification can easily be performed. One could consider the reservoir as a spatiotemporal kernel, in which the mapping to a high-dimensional space is computed explicitly. In this letter, we build on this idea and extend the concept of ESNs to infinite-sized recurrent neural networks, which can be considered recursive kernels that subsequently can be used to create recursive support vector machines. We present the theoretical framework, provide several practical examples of recursive kernels, and apply them to typical temporal tasks."

"The predictability of errors in deterministic temperature forecasts is investigated. More precisely, the aim is to issue warnings whenever the differences between forecast and verification exceed a given threshold. The warnings are generated by analyzing the output of an ensemble forecast system in terms of a decision making approach. The quality of the resulting predictions is evaluated by computing receiver operating characteristics, the Brier score, and the Ignorance score. Special emphasis is also given to the question whether rare events are better predictable."

"This article reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory (ergodic theory). In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this article clarifies von Neumann’s contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the article is that von Neumann’s argument described in his letter to Ulam is the very first proof that spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic."

"We present an efficient method for estimating variables and parameters of a given system of ordinary differential equations by adapting the model output to an observed time series from the (physical) process described by the model. The proposed method is based on (unconstrained) nonlinear optimization exploiting the particular structure of the relevant cost function. To illustrate the features and performance of the method, simulations are presented using chaotic time series generated by the Colpitts oscillator, the three-dimensional Hindmarsh-Rose neuron model, and a nine-dimensional extended Rössler system." --- Sounds like Hooker & Ramsay.

To many math symbols to copy the abstract. Shorter: iterating randomly chosen Lipschitz maps can lead to time-averges converging to a heavy-tailed distribution.

"Let $(U_n(t))_{tinR^d}$ be the empirical process associated to an $R^d$-valued stationary process $(X_i)_{ige 0}$. We give general conditions, which only involve processes $(f(X_i))_{ige 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{tinR^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications."

"Originally published in 1981" - still relevant?

"We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting, if the one-dimensional (1-D) distribution of the samples is comparable to a known reference distribution, and the noise is independent of the samples and has known mixing rates. The estimates are applied to deterministic sampling schemes, in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector, and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories"

"A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas."

"In this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements. These statistics can be evaluated on any reconstructed attractor, thereby allowing a direct comparison among different approaches: (uniform or nonuniform) delay vectors, PCA, Legendre coordinates, etc. It can also be used to select the most appropriate parameters of a reconstruction strategy. In the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function. Its definition is based on theoretical arguments on noise amplification, the complexity of the reconstructed attractor, and a direct measure of local stretch which constitutes an irrelevance measure. The proposed method is demonstrated on synthetic and experimental time series."

"V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers."

"We study the impact of stochastic perturbations to deterministic dynamical systems using the formalism of the Ruelle response theory. We find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the case of additive and multiplicative noise. "

"In this Letter we show that the analysis of Lyapunov-exponents fluctuations contributes to deepen our understanding of high-dimensional chaos. This is achieved by introducing a Gaussian approximation for the entropy function that quantifies the fluctuation probability. More precisely, a diffusion matrix D (a dynamical invariant itself) is measured and analysed in terms of its principal components. The application of this method to four (conservative, as well as dissipative) models, allows: (i) quantifying the strength of the effective interactions among the different degrees of freedom; (ii) unveiling microscopic constraints such as those associated to a symplectic structure; (iii) checking the hyperbolicity of the dynamics."

"We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of the theory of dynamics, as invariant measures and invariant sets, showing that even if they can be computed with arbitrary precision in many interesting cases, there exists some cases in which they can not. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem)."

I don't see how what they are promising could possibly be true in the generality which they also promise, but "to be shot after a fair trial".

" We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity."

"In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. "

"This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included"

library has this but in off-site storage; request.
"There are many ways of introducing the concept of probability in classical, deterministic physics. This volume is concerned with one approach, known as 'the method of arbitrary functions', which was first considered by Poincare. ... proceeds by associating some uncertainty to our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. By modeling this uncertainty by a probability density distribution ... analyze how the state of the system evolves through time. ... examples as diverse as bouncing balls, simple and coupled harmonic oscillators, integrable systems (such as spinning tops), planetary motion, and billiards. ... study the speed of convergence for solutions in order to determine the practical relevance of the method of arbitrary functions for specific examples. ... new results on convergence, and tractable upper bounds are derived"

"probabilities as deriving from ranges in suitably structured initial state spaces. Roughly, the probability of an event is the proportion of initial states that lead to this event in the space of all possible initial states, provided that this proportion is approximately the same in any not too small interval of the initial state space. This idea can also be expressed by saying that in the types of situations that give rise to probabilistic phenomena we may expect to find an initial state space such that any "reasonable" density function over this space leads to the same probabilities for the possible outcomes"

"We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem."

"It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincar'e returns. The close relation between periodic orbits and the Poincar'e returns allows for analytical and semi-analytical estimations of relevant quantities in dynamical systems, as the decay of correlation and the Kolmogorov-Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach is highly oriented to the treatment of experimental systems."

"The guiding question of this paper is: how similar are deterministic descriptions and indeterministic descriptions from a predictive viewpoint? The deterministic and indeterministic descriptions of concern in this paper are measure-theoretic deterministic systems and stochastic processes, respectively. I will explain intuitively some mathematical results which show that measure-theoretic deterministic systems and stochastic processes give more often the same predictions than one might perhaps have expected, and hence that from a predictive viewpoint these descriptions are quite similar." This needs saying?!?

"This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem."

Sounds _very much_ like the old Fox & Keizer papers (PRL 64 [1990]: 249 and PRA 43 [1991]: 1709) on "amplification of intrinsic molecular fluctuations by chaos"; but those aren't cited. Read carefully; write a comment?

Some variant of particle filtering with state augmentation?

Shorter Prof. Dr. Peter Grassberger; Tsallis is teh suxx0r!

Looks cool.

"We claim that looking at probability distributions of emph{finite time} largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincar'e recurrences in the -quite delicate- case of dynamical systems with weak chaotic properties."

... the unfortunate Strogatz column is made worse by the fact that there are actual dynamical models of marriage, with at least some connection to empirical data, rather than being derived _ex ano_.

This is really not good. Strogatz knows much better --- why is he doing this?

Favorable review in J. Stat. Phys. (doi: s10955-009-9712-6)

But the deadline is February 1st! Why am I only hearing about this now? Wah.

"replicator equations (RE) are among the basic tools in mathematical theory of selection and evolution. We develop a method for reducing a wide class of the RE, which in general are systems of differential equations in Banach space to escort systems of ODEs that in many cases can be explored analytically. The method has potential for different applications; some examples are given."

- The method does not seem to apply when fitness fluctuates stochastically.

How much information (in the Fisher sense) does the present state of a recurrent dynamical network retain about the history of its inputs? All, or almost all, done for linear-Gaussian systems, but numerical results for nonlinear, non-Gaussian systems would be straightforward in principle.

Andy's book is appearing in print at last. (SIAM seems to indicate it's available now, the usual online bookstores say not until the end of the year.)

Vancouver in June.

See http://bactra.org/reviews/computational-beauty-of-nature/

Full text of book published by Springer, sans snazy formatting. Looks non-crazy based on the preface and table of contents.

Unquestionably still the best book on the mathematical foundations of complex systems theory, and the measurement of complexity.

"Unified Growth Theory", i.e., economic growth. Talks a lot about attractors. Anything to this?

Online book (PDF). Looks very pure-mathy.

Staring Marlowe the Cat

Excellent introduction to the nonlinear-dynamics approach to time series