Random matrix ensembles in hyperchaotic classical dissipative dynamic systems

Abstract

We study the statistical fluctuations of Lyapunov exponents in the discrete version of the non-integrable perturbed sine-Gordon equation, the dissipative AC- and DC-driven Frenkel–Kontorova model. Our analysis shows that the fluctuations of the exponent spacings in the strictly overdamped limit, which is nonchaotic, conform to an uncorrelated Poisson distribution. By studying the spatiotemporal dynamics, we relate the emergence of the Poissonian statistics to Middleton’s no-passing rule. Next, by scanning values of the DC drive and the particle mass, we identify several parameter regions for which this one-dimensional model exhibits hyperchaotic behavior. Furthermore, in the hyperchaotic regime where roughly fifty percent of the exponents are positive, the fluctuations exhibit features of the correlated universal statistics of the Gaussian orthogonal ensemble (GOE). Due to the dissipative nature of the dynamics, we find that the match between the Lyapunov spectrum statistics and the universal statistics of GOE is not complete. Finally, we present evidence supporting the existence of the Tracy–Widom distribution in the fluctuation statistics of the largest Lyapunov exponent.