A Chaotic Limit Cycle Paradox

Duffing’s equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achiev-able for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent which, in 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