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Multiresonant control of two-dimensional dynamical systems.

I Barth1, L Friedland

  • 1Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 7, 2007
PubMed
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Continuous phase-locking, or double autoresonance, controls two degree of freedom (2D) nonlinear dynamical systems using external perturbations. This method enables stable control and formation of nearly periodic trajectories in various driven systems.

Area of Science:

  • Nonlinear Dynamics
  • Control Theory
  • Chaos Theory

Background:

  • Nonlinear dynamical systems with two degrees of freedom (2D) present complex behaviors.
  • Controlling such systems often requires sophisticated methods.
  • External perturbations can influence system dynamics.

Purpose of the Study:

  • To demonstrate control of 2D nonlinear dynamical systems via double autoresonance.
  • To establish conditions for the stability of 2D autoresonance.
  • To explore applications in forming nearly periodic trajectories.

Main Methods:

  • Utilizing continuous phase-locking between system variables and external oscillating perturbations.
  • Analyzing stability using the Hessian matrix elements of the unperturbed system.

Related Experiment Videos

  • Investigating system response to simultaneous passage through two linear resonances.

    • Main Results:

    • Identified conditions for stable 2D autoresonance and classified doubly autoresonant solutions.
    • Determined that threshold driving amplitudes scale as alpha(3/4) for accessing autoresonant states.
    • Showcased formation of nearly periodic trajectories in driven 2D systems.

    Conclusions:

    • Double autoresonance offers a viable control strategy for 2D nonlinear dynamical systems.
    • The method is applicable to systems with stable equilibria, leading to predictable trajectories.
    • Demonstrated effectiveness in diverse systems like Toda chains and double-well potentials.