The high forecasting complexity of stochastically perturbed periodic orbits limits the ability to distinguish them from
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ORIGINAL PAPER
The high forecasting complexity of stochastically perturbed periodic orbits limits the ability to distinguish them from chaos Navendu S. Patil
· Joseph P. Cusumano
Received: 28 February 2020 / Accepted: 25 August 2020 © Springer Nature B.V. 2020
Abstract A long-standing question in applications of dynamical systems theory is how to distinguish noisy signals from chaotic steady states. Informationtheoretic 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 N. S. Patil (B) · J. P. Cusumano Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA e-mail: [email protected] J. P. Cusumano e-mail: [email protected] N. S. Patil Present address: Department of Kinesiology, Pennsylvania State University, University Park, PA 16802, USA
difficult to distinguish it from chaotic states at adjacent parameters, even with small noise. However, for a lowperiod ( 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. Keywords Statistical complexity · Entropy rate · Generating partitions · Noisy chaos · Nonlinear oscillators · Causal state models
1 Introduction Experimental nonlinear dynamics is concerned with characterizing nonlinear systems, including those evolving in high-dimensional phase spaces, through analysis of noisy time-series measurements [18]. A phenomenological understanding of such systems is also possible through dynamic simulations of first-principle models. A fundamental issue that has received considerable attention in the literature [7,13,24,29,32,37, and references therein] is distinguishing noisy signals from chaotic steady-state trajectories. While qualita-
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tively distinct dynam
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