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19
On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
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Cited by 39 (11 self)
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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Recent Results About Stable Ergodicity
 In Smooth ergodic theory and its applications
, 2000
"... this paper, has been directed toward extending their results beyond Axiom A. ..."
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Cited by 16 (3 self)
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this paper, has been directed toward extending their results beyond Axiom A.
Pathological Foliations and Removable Zero Exponents
, 1999
"... this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of dieomorphisms. As a curious byproduct of our construction, we prove that a pathological feature of central foliations { the complete failure of absolute continuity { can exist in a ..."
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Cited by 16 (4 self)
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this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of dieomorphisms. As a curious byproduct of our construction, we prove that a pathological feature of central foliations { the complete failure of absolute continuity { can exist in a
Frontiers in complex dynamics
 Bull. of Amer. Math. Soc
"... Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complexanalytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The ra ..."
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Cited by 8 (2 self)
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Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complexanalytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given
Basic Types of CoarseGraining
, 2006
"... We consider two basic types of coarsegraining: the Ehrenfest’s coarsegraining and its extension to a general principle of nonequilibrium thermodynamics, and the coarsegraining based on uncertainty of dynamical models and εmotions (orbits). Nontechnical discussion of basic notions and main coa ..."
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Cited by 6 (3 self)
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We consider two basic types of coarsegraining: the Ehrenfest’s coarsegraining and its extension to a general principle of nonequilibrium thermodynamics, and the coarsegraining based on uncertainty of dynamical models and εmotions (orbits). Nontechnical discussion of basic notions and main coarsegraining theorems are presented: the theorem about entropy overproduction for the Ehrenfest’s coarsegraining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for εmotions of general dynamical systems including structurally unstable systems. A brief discussion of two other types, coarsegraining by rounding and by small noise, is also presented. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfest’s coarsegraining.
Perturbation Theory of Dynamical Systems
, 2001
"... Please send any comments to berglund@math.ethz.ch Zurich, November 2001 3 4 Contents 1 Introduction and Examples 1 1.1 OneDimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Si ..."
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Cited by 3 (0 self)
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Please send any comments to berglund@math.ethz.ch Zurich, November 2001 3 4 Contents 1 Introduction and Examples 1 1.1 OneDimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Singular Perturbations: The Van der Pol Oscillator . . . . . . . . . . . . . . 10 2 Bifurcations and Unfolding 13 2.1 Invariant Sets of Planar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 The PoincareBendixson Theorem . . . . . . . . . . . . . . . . . . . 21 2.2 Structurally Stable Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Definition of Structural Stability . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Peixot
GradientLike Dynamics in Neural Networks
, 1993
"... This report presents a formalism that enables the dynamics of a broad class of neural networks to be understood. A number of previous works have analyzed the Lyapunov stability of neural network models. This type of analysis shows that the excursion of the solutions from a stable point is bounded. T ..."
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Cited by 1 (1 self)
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This report presents a formalism that enables the dynamics of a broad class of neural networks to be understood. A number of previous works have analyzed the Lyapunov stability of neural network models. This type of analysis shows that the excursion of the solutions from a stable point is bounded. The purpose of this work is to present a model of the dynamics that also describes the phase space behavior as well as the structural stability of the system. This is achieved by writing the general equations of the neural network dynamics as a gradientlike system. In this paper some important properties of gradientlike systems are developed and then it is demonstrated that a broad class of neural network models are expressible in this form. y y Acknowledgments: This research was supported by a grant from Boeing Computer Services under Contract W300445. 1 Chapter 1 Introduction In this paper we propose a formalism that allows three critical issues in the study of unsupervised neural n...
Riemann surfaces, dynamics and geometry Course Notes Harvard University — Math 275
"... 1.1 Examples of hyperbolic manifolds............. 3 1.2 Examples of rational maps................. 10 ..."
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1.1 Examples of hyperbolic manifolds............. 3 1.2 Examples of rational maps................. 10