Pseudo-deterministic Chaotic Systems

Celso Grebogi, R. L. Viana, S. E. S. Pinto, J. R. R. Barbosa

Research output: Contribution to journalArticle

16 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)3235-3253
Number of pages18
JournalInternational Journal of Bifurcation and Chaos
Volume13
Publication statusPublished - 2003

Keywords

  • chaotic systems
  • hyperbolic systems
  • shadowing
  • Lyapunov
  • UNSTABLE DIMENSION VARIABILITY
  • LYAPUNOV EXPONENTS
  • PERIODIC-ORBITS
  • DYNAMIC-SYSTEMS
  • ATTRACTORS
  • SYNCHRONIZATION
  • SETS
  • TRAJECTORIES
  • BIFURCATION
  • OSCILLATORS

Cite this

Grebogi, C., Viana, R. L., Pinto, S. E. S., & Barbosa, J. R. R. (2003). Pseudo-deterministic Chaotic Systems. International Journal of Bifurcation and Chaos, 13, 3235-3253.

Pseudo-deterministic Chaotic Systems. / Grebogi, Celso; Viana, R. L.; Pinto, S. E. S.; Barbosa, J. R. R.

In: International Journal of Bifurcation and Chaos, Vol. 13, 2003, p. 3235-3253.

Research output: Contribution to journalArticle

Grebogi, C, Viana, RL, Pinto, SES & Barbosa, JRR 2003, 'Pseudo-deterministic Chaotic Systems', International Journal of Bifurcation and Chaos, vol. 13, pp. 3235-3253.
Grebogi, Celso ; Viana, R. L. ; Pinto, S. E. S. ; Barbosa, J. R. R. / Pseudo-deterministic Chaotic Systems. In: International Journal of Bifurcation and Chaos. 2003 ; Vol. 13. pp. 3235-3253.
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AU - Grebogi, Celso

AU - Viana, R. L.

AU - Pinto, S. E. S.

AU - Barbosa, J. R. R.

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N2 - 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.

AB - 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.

KW - chaotic systems

KW - hyperbolic systems

KW - shadowing

KW - Lyapunov

KW - UNSTABLE DIMENSION VARIABILITY

KW - LYAPUNOV EXPONENTS

KW - PERIODIC-ORBITS

KW - DYNAMIC-SYSTEMS

KW - ATTRACTORS

KW - SYNCHRONIZATION

KW - SETS

KW - TRAJECTORIES

KW - BIFURCATION

KW - OSCILLATORS

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JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

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