A long-standing question in applications of dynamical systems theory is how to distinguish noisy signals from chaotic steady states. Information-theoretic measures hold promise to resolve this problem. We apply two such measures to numerically computed phase-space trajectories of continuous-state nonlinear oscillators: forecasting or statistical complexity, which quantifies the minimum memory required for the optimal prediction of discrete observables, and the entropy rate, which quantifies their intrinsic unpredictability. We estimate empirical generating partitions to obtain discrete observables faithfully representing continuous-state chaotic time series. We focus on the problem of distinguishing stochastically perturbed periodic orbits from chaotic attractors that exist at nearby parameter values, in a region of the parameter space where a strange invariant set exists. We find that a stochastically perturbed, stable, high-period (\(p=15\)) orbit of a periodically driven Duffing oscillator admits high values of both information measures, making it difficult to distinguish it from chaotic states at adjacent parameters, even with small noise. However, for a low-period (\(p=3\)) orbit, such a distinction becomes easier, as both measures admit considerably lower values compared to a chaotic attractor at a nearby parameter. Furthermore, the forecasting complexity of the selected periodic orbits increases with noise as they undergo a transition to “noise-induced chaos.” For sufficiently high noise levels, our ability to distinguish chaos from noise depends on model-order selection when estimating forecasting complexity and also on the exact choice of discrete observables used to encode phase-space trajectories.

在动力系统理论的应用中，一个长期存在的问题是如何区分噪声信号和混沌稳态。信息理论方法有望解决这个问题。我们对连续状态非线性振荡器的数值计算的相空间轨迹应用了两种这样的措施：预测或统计复杂性，量化了离散可观测物最佳预测所需的最小内存;熵率，量化了其固有的不可预测性。我们估计经验生成分区，以获得忠实地代表连续状态混沌时间序列的离散可观测值。我们关注于将随机扰动的周期性轨道与附近参数值处存在的混沌吸引子区分开来的问题，在存在奇怪的不变集的参数空间区域中。我们发现，随机扰动，稳定，高周期（周期性驱动的Duffing振荡器的\（p = 15 \））轨道允许两个信息量度都很高，即使在噪声很小的情况下，也很难将其与相邻参数的混沌状态区分开。但是，对于低周期（\（p = 3 \））轨道，这样的区分变得更加容易，因为与附近参数处的混沌吸引子相比，这两种措施都允许其值大大降低。此外，选定的周期性轨道的预测复杂性会随着噪声向“噪声引起的混沌”的转变而随噪声而增加。对于足够高的噪声水平，我们区分混沌与噪声的能力取决于估计预测复杂度时的模型顺序选择，还取决于用于对相空间轨迹进行编码的离散可观观测值的准确选择。