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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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Plykin-type attractor in nonautonomous coupled oscillators.

Sergey P Kuznetsov¹

¹Kotel'nikov's Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Saratov, Russian Federation.

Chaos (Woodbury, N.Y.)

|April 2, 2009

Summary

This study analyzes coupled nonautonomous oscillators, revealing a Plykin-type attractor with hyperbolic properties. Numerical simulations explore complex dynamics, including Lyapunov exponents and power spectral density.

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Area of Science:

Nonlinear Dynamics
Chaos Theory
Oscillator Systems

Background:

Coupled oscillators exhibit complex behaviors.
Nonautonomous systems present unique dynamic challenges.
Understanding attractors is key to characterizing system stability.

Purpose of the Study:

To investigate the dynamics of two coupled nonautonomous oscillators.
To derive and analyze the Poincare map for the system.
To confirm the hyperbolic nature of the system's attractor.

Main Methods:

Formulating differential equations for complex amplitudes.
Deriving an explicit Poincare map.
Reducing the map to a 3D representation.
Utilizing the cone criterion for hyperbolic verification.
Performing numerical simulations.

Main Results:

An explicit Poincare map was derived and reduced to 3D.
A Plykin-type attractor was identified on an invariant sphere.
The cone criterion confirmed the attractor's hyperbolic nature.
Numerical studies presented attractor portraits, Lyapunov exponents, and power spectral density.

Conclusions:

The coupled nonautonomous oscillator system exhibits complex, hyperbolic chaotic dynamics.
The derived 3D Poincare map effectively characterizes the system's long-term behavior.
Numerical results provide insights into the spectral properties and stability of the chaotic attractor.