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.

我们研究了不可积扰动正弦-戈登方程（耗散交流和直流驱动的 Frenkel-Kontorova 模型）的离散版本中李雅普诺夫指数的统计波动。我们的分析表明，严格过阻尼极限内指数间距的波动是非混沌的，符合不相关的泊松分布。通过研究时空动态，我们将泊松统计的出现与米德尔顿的不通过规则联系起来。接下来，通过扫描直流驱动和粒子质量的值，我们确定了这个一维模型表现出超混沌行为的几个参数区域。此外，在大约 50% 的指数为正的超混沌状态中，波动表现出高斯正交系综（GOE）的相关通用统计。由于动力学的耗散性质，我们发现 Lyapunov 谱统计量与 GOE 的通用统计量之间的匹配不完整。最后，我们提供了支持最大李雅普诺夫指数波动统计中存在 Tracy-Widom 分布的证据。