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Geometric phase for nonlinear oscillators from perturbative renormalization group.

D A Khromov1, M S Kryvoruchko2, D A Pesin3

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We present a renormalization-group method for nonlinear oscillators, applicable to both constant and changing parameters. This approach determines the geometric phase acquired by oscillators with varying parameters.

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Area of Science:

  • Nonlinear Dynamics
  • Theoretical Physics
  • Mathematical Physics

Background:

  • Nonlinear oscillators are fundamental in various scientific fields.
  • Existing methods may not fully capture dynamics with time-varying parameters.
  • Understanding parameter evolution effects is crucial for system analysis.

Purpose of the Study:

  • To develop a novel renormalization-group (RG) approach for general nonlinear oscillator problems.
  • To extend the RG method to handle both autonomous and nonautonomous systems.
  • To utilize the RG framework for quantifying geometric phase in parameter-varying oscillators.

Main Methods:

  • Formulation of an RG approach based on the exact group law of ordinary differential equations.
  • Application to autonomous models with time-independent parameters.
  • Extension to nonautonomous models with slowly varying parameters.

Main Results:

  • The RG equations provide a systematic way to analyze nonlinear oscillators.
  • The developed method accurately captures dynamics for both autonomous and nonautonomous cases.
  • The RG framework successfully determines the geometric phase acquired during parameter changes.

Conclusions:

  • The proposed RG approach offers a powerful tool for studying nonlinear oscillators.
  • This method provides new insights into the behavior of systems with time-varying parameters.
  • The application to Van der Pol and Van der Pol-Duffing models demonstrates the method's efficacy.