Lyapunov dimension

Chaotic, Forced, and Coupled Oscillators

Published in LM Pismen, Working with Dynamical Systems, 2020

The geometrical and dynamic measures are commonly related. Kaplan and Yorke (1979) introduced “Lyapunov dimension” dependent on the number of positive Lyapunov exponents, claiming that it is equal to the information dimension for “typical” systems. This might be vaguely true, but the case of SNA certifies that nontypical systems do exist as well.

A study of the nonlinear response and chaos suppression in a magnetically levitated system

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Journal Information

Published in Australian Journal of Mechanical Engineering, 2020

Shun-Chang Chang, Yeou-Feng Lue

where j is the largest integer that satisfies . This technique yields a Lyapunov dimension for Equation (3) of Ω = 10.0 rad/s with dL = 1. Since the Lyapunov dimension is an integer, the system exhibits periodic motion. When the parameter Ω increases beyond the bifurcation point (e.g. Ω = 25.0 rad/s), the Lyapunov exponents are λ1 = 1.3711, λ2 = −5.5069 and λ3 = −140.1338, and the Lyapunov dimension is dL = 1.25. Because the Lyapunov dimension is not an integer, the system is shown to exhibit chaotic motion. Accordingly, for periodic solutions, the Lyapunov dimension takes an integer value; for strange attractors, the Lyapunov dimension may take a non-integer value.

Lyapunov dimension

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Chaotic, Forced, and Coupled Oscillators

A study of the nonlinear response and chaos suppression in a magnetically levitated system