The high forecasting complexity of stochastically perturbed periodic orbits limits the ability to distinguish them from chaos

Abstract

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.

Appendix A: A custom SRK scheme for the numerical integration of the SDEs with additive noise

Our custom six-stage SRK scheme of global (stochastic) order 1.5 and deterministic order 5 yields a discrete approximation, ${{\varvec{y}}}_1^M {:}{=}\{{{\varvec{y}}}_r: r = 1,2,\ldots ,M\}$ at times $\{t_r = rh\}$, of the solution trajectory of the SDEs (1) with constant additive noise (i.e., ${{\varvec{g}}}=\text {constant}$) given by:

$$\begin{aligned} \displaystyle {{\varvec{y}}}_{r+1} = {{\varvec{y}}}_r + \sum _{i=1}^{6} \alpha _i h \, {{\varvec{f}}}(t_r+c_i h, H_i) +{{\varvec{g}}}\, I_{(1)}, \end{aligned}$$

(19)

Table 1 Cash–Karp parameters [6, 30], $\{{{\varvec{c}}}, {{\varvec{A}}}, {{\varvec{\alpha }}}\}$ for the six-stage explicit Runge–Kutta scheme of order 5 for the ODEs

with stages, for $i = 1,2,\ldots ,6$,

$$\begin{aligned} \displaystyle H_i = {{\varvec{y}}}_{r} + \sum _{j=1}^{6} A_{ij} h \, {{\varvec{f}}}(t_r+c_j h, H_j) + {{\varvec{g}}} \frac{I_{(1,0)}}{h} \, (Be)_{i},\nonumber \\ \end{aligned}$$

(20)

where the method coefficients: ${{\varvec{c}}}$, ${{\varvec{\alpha }}}$, and ${{\varvec{A}}}$, are from Table 1; the 6-by-1 coefficient vector ${{\varvec{B}}e}$ is in Table 2; ${{\varvec{y}}}_0 {:}{=}{{\varvec{x}}}(0)$; and h is the time step. The random variables, $I_{(1)}$ and $I_{(1,0)}$, are the Itô stochastic integrals, exactly modeled [19, 23] as:

$$\begin{aligned} \displaystyle I_{(1)} = \sqrt{h}\, \zeta _1,\quad I_{(1,0)} = \frac{1}{2} h^{3/2} \left( \zeta _1 + \frac{1}{\sqrt{3}} \zeta _2 \right) \end{aligned}$$

(21)

where $\zeta _1, \zeta _2$ are independent standard normal random variables. The iterate of the Poincaré map ${{\varvec{F}}}_{\pi }^{\sigma }$ (Eq. 3) at the end of the $n^{\text {th}}$ half-period of the forcing is obtained as: ${{\varvec{x}}}_n=-{{\varvec{y}}}_M$, where ${{\varvec{y}}}_M$ is found iteratively using Eq. (19).

Table 2 Coefficient vector ${{\varvec{B}}e}$ derived in [26] for a custom SRK scheme (Eqs. 19, 20) of order 1.5, starting from a general scheme in [31]