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Easy-to-implement method to target nonlinear systems.

Murilo S. Baptista1, Ibere L. Caldas

  • 1Institute of Physics, University of Sao Paulo, C. P. 66318, CEP 05315-970 Sao Paulo, S.P., Brazil.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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This study introduces a novel method for rapidly controlling chaotic systems to a target state using minimal control perturbations. The technique is demonstrated on various chaotic models, showing its effectiveness and robustness.

Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Control Theory

Background:

  • Controlling chaotic systems to a specific target state is a significant challenge in nonlinear dynamics.
  • Existing methods often require complex control strategies or detailed system models.

Purpose of the Study:

  • To develop a rapid and efficient method for directing chaotic systems to a target state.
  • To apply and validate this method on diverse chaotic systems, including maps and circuits.
  • To enhance existing control techniques by reducing reliance on low-dimensional mappings.

Main Methods:

  • A control perturbation strategy using a few distinct amplitudes was employed.
  • The method was applied to the one-dimensional Logistic map, the two-dimensional Henon map, and the three-dimensional Double Scroll circuit.

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  • Numerical simulations were used to analyze trajectories, stability, and robustness.

    • Main Results:

    • The method successfully directed chaotic systems to target states across different dimensions.
    • For the Logistic map, trajectories were shown to follow a stable manifold of the target.
    • The Henon map demonstrated the creation and stabilization of unstable periodic orbits, with noise robustness verified.

    Conclusions:

    • The presented method offers an effective way to control chaotic dynamics with minimal interventions.
    • It provides an advancement over previous methods by not requiring low-dimensional mappings for complex systems like the Double Scroll circuit.
    • The technique shows promise for applications in various fields involving chaotic behavior.