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PSEUDO-DETERMINISTIC CHAOTIC SYSTEMS

R. L. VIANA

Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19081, 81531-990, Curitiba, Paraná, Brasil

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S. E. DE S. PINTO

Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19081, 81531-990, Curitiba, Paraná, Brasil

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J. R. R. BARBOSA

Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19081, 81531-990, Curitiba, Paraná, Brasil

Permanent Address: Departamento de Matemática, Universidade Federal do Paraná, 81531-990, Curitiba, PR, Brasil.

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, and

C. GREBOGI

Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970, São Paulo, São Paulo, Brasil

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https://doi.org/10.1142/S0218127403008636Cited by:21 (Source: Crossref)

Abstract

We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

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