A Choatic Limit Cycle Paradox
A Chaotic Limit Cycle Paradox1
Walter J. Grantham
Byoungsoo Lee2
Department of Mechanical and Materials Engineering
Washington State University
Pullman, Washington 99164-2920
Abstract
Duffing's equation with sinusoidal forcing produces chaos for
certain combinations of the forcing amplitude and frequency. To determine
the most chaotic response achievable for given bounds on the input force, an
optimal control problem was investigated to maximize the largest Lyapunov
exponent which, it this case, also corresponds to maximizing the
Kaplan-Yorke Lyapunov fractal dimension. The resulting bang-bang optimal
feedback controller yielded a bounded attractor with a positive largest
Lyapunov exponent and a fractional Lyapunov dimension, indicating a chaotic
strange attractor. Indeed, the largest Lyapunov exponent was approximately
twice as large as that achieved with sinusoidal forcing at the same
amplitude. However, the resulting attractor is just a stable limit cycle and
is not chaotic or fractal at all! This contradicts the basic idea that a
bounded attractor with at least one positive Lyapunov exponent must be
chaotic and fractal.
This paper provides details of this chaotic limit cycle paradox
and the resolution of the paradox. In particular, for systems of
differential equations with only piecewise differentiable right-hand sides,
a jump discontinuity condition must be imposed on the state perturbations in
order to compute correct Lyapunov exponents.
Keywords: Chaos, Lyapunov exponents, discontinuities,
perturbations, jump conditions.
1 -- Grantham, W.J. and Lee, B., "A Chaotic Limit Cycle Paradox,"
Dynamics and Control, Vol. 3, No. 2, April 1993, pp. 157-171.
2 -- Presently at: Department of Mechanical Design, College of Engineering, Kei
Myung University, Dal Seo Gu, Shin Dang Dong 1000, Taegu 704-701, Korea.
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