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Complex dynamics in simple systems with periodic parameter oscillations.

L Héctor Juárez1, Holger Kantz, Oscar Martínez

  • 1Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Apdo. Postal 55-534, 09340 México DF, Mexico.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
PubMed
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Complex periodic orbits emerge in systems with oscillating parameters, even within chaotic regimes. These intricate patterns, characterized by Lyapunov exponents, are found in physical systems like fluid flow and dripping faucets.

Area of Science:

  • Nonlinear dynamics and chaos theory
  • Complex systems analysis
  • Statistical mechanics

Background:

  • Systems with periodically oscillating parameters can exhibit complex behaviors, including periodic and nonperiodic orbits.
  • Ergodic averages, such as Lyapunov exponents, are crucial for characterizing the stability and nature of these orbits in the long-time limit.
  • A negative maximal Lyapunov exponent indicates a stable periodic orbit, while positive exponents suggest chaotic dynamics.

Purpose of the Study:

  • To investigate the emergence and characteristics of complex periodic orbits in systems with oscillating parameters.
  • To explore the relationship between parameter values supporting periodic motion and those leading to chaotic behavior.
  • To understand the impact of noise on the distinguishability of dynamics with different Lyapunov exponents.

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Main Methods:

  • Analysis of systems with periodically oscillating parameters in the long-time limit.
  • Calculation of ergodic averages, specifically maximal Lyapunov exponents, to classify orbital stability.
  • Application of the epsilon-uncertain points technique to map parameter spaces.

Main Results:

  • Extremely complicated periodic orbits, featuring both contracting and expanding phases, naturally arise.
  • Parameter values supporting these complex periodic motions are densely embedded within the chaotic parameter regime.
  • The introduction of even small amounts of noise makes dynamics with positive and negative Lyapunov exponents indistinguishable.

Conclusions:

  • Complex periodic orbits are an inherent feature of periodically driven systems and are intricately linked with chaotic dynamics.
  • The epsilon-uncertain points method reveals a dense coexistence of order and chaos in the parameter space.
  • Noise significantly impacts the observable dynamics, blurring the lines between stable and chaotic behaviors in physical systems.