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Characterizing the dynamics of higher dimensional nonintegrable conservative systems.

Cesar Manchein¹, Marcus W Beims, Jan M Rost

¹Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Brazil.

Chaos (Woodbury, N.Y.)

|October 2, 2012

Summary

This study characterizes phase space dynamics in high-dimensional systems using sticky motion and finite time Lyapunov exponents (FTLEs). Sticky motion helps distinguish mixed from chaotic regimes in conservative maps.

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Area of Science:

Nonlinear dynamics
Statistical mechanics
Chaos theory

Background:

Phase space dynamics in higher dimensional nonintegrable conservative systems are complex.
Chaotic trajectories can exhibit

Purpose of the Study:

To characterize phase space dynamics in higher dimensional nonintegrable conservative systems.
To investigate the effect of sticky motion on finite time Lyapunov exponents (FTLEs) distribution.
To differentiate between mixed and totally chaotic regimes.

Main Methods:

Analysis of sticky motion's effect on finite time Lyapunov exponents (FTLEs) distribution.
Utilizing four distinct measures related to positive FTLEs distributions.
Systematic study of conservative maps from 2D to 20D.

Main Results:

Sticky motion was detected in all unstable directions for high dimensional cases (d=10, 20) above a nonlinearity parameter threshold K(d).
A clear transition from mixed to totally chaotic motion was identified as K increased, occurring simultaneously in all unstable directions.
All four statistical measures effectively characterized motion in high dimensional systems.

Conclusions:

Sticky motion is a key indicator for characterizing phase space dynamics in high-dimensional conservative systems.
Finite time Lyapunov exponents (FTLEs) distributions provide sensitive measures for detecting and differentiating chaotic regimes.
The study successfully identified transitions between mixed and chaotic dynamics across various dimensions.