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

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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.

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  1. Considering \(\delta \in [0.2,0.6]\) as a bifurcation parameter, we also estimated \(h_{\mu }\) values for about 300 chaotic attractors using the CSSR algorithm (\(N=2\times 10^6\)), as done here for \(\delta =0.25\): \(h_{\mu }\) values were higher than \(\lambda _1\) by \(\approx 4 \%\) (averaged over all \(\delta \) parameters) [26].


  1. Abad, A., Barrio, R., Blesa, F., Rodríguez, M.: Algorithm 924: TIDES, a Taylor series integrator for differential equations. ACM Trans. Math. Softw. 39(1), 1–28 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  2. Ash, R.B.: Information Theory. Dover, New York (1965)

    MATH  Google Scholar 

  3. Badii, R., Politi, A.: Complexity: Hierarchical Structures and Scaling in Physics. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  4. Barreira, L., Pesin, Y.: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  5. Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965)

    MATH  Google Scholar 

  6. Cash, J.R., Karp, A.H.: A variable order Runge–Kutta method for initial value problems with rapidly varying right-hand sides. ACM Trans. Math. Softw. 16(3), 201–222 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  7. Cencini, M., Falcioni, M., Olbrich, E., Kantz, H., Vulpiani, A.: Chaos or noise: difficulties of a distinction. Phys. Rev. E 62(1), 427–437 (2000).

    Article  Google Scholar 

  8. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Interscience, Hoboken (1991)

    Book  Google Scholar 

  9. Crutchfield, J.P.: The calculi of emergence: computation, dynamics and induction. Physica D 75(1–3), 11–54 (1994).

    Article  MATH  Google Scholar 

  10. Crutchfield, J.P., Young, K.: Inferring statistical complexity. Phys. Rev. Lett. 63(2), 105–108 (1989).

    Article  MathSciNet  Google Scholar 

  11. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 617–656 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  12. Fraser, A.M.: Hidden Markov Models and Dynamical Systems. SIAM Press, Philadelphia (2008)

    Book  Google Scholar 

  13. Gao, J., Cao, Y., Tung, W., Hu, J.: Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond. Wiley, Hoboken (2007)

    Book  Google Scholar 

  14. Grassberger, P.: Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys. 25(9), 907–938 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  15. Haslinger, R., Klinkner, K.L., Shalizi, C.R.: The computational structure of spike trains. Neural Comput. 22(1), 121–157 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. Chaos 9(2), 413–435 (1999).

    Article  MATH  Google Scholar 

  17. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  18. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  19. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, Stochastic Modelling and Applied Probability, vol. 23. Springer, Berlin (1992)

    Book  Google Scholar 

  20. Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inf. Theory 22(1), 75–81 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  21. Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation, 2nd edn. Prentice-Hall, Upper Saddle River (1997)

    Google Scholar 

  22. Marton, K., Shields, P.C.: Entropy and the consistent estimation of joint distributions. Ann. Probab. 22(2), 960–977 (1994). Correction. Ann. Probab. 24(1), 541–545. (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Scientific Computation. Springer, Berlin (2004)

    Book  Google Scholar 

  24. Olivares, F., Plastino, A., Rosso, O.A.: Contrasting chaos with noise via local versus global information quantifiers. Phys. Lett. A 376(19), 1577–1583 (2012).

    Article  MATH  Google Scholar 

  25. Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002).

    Book  MATH  Google Scholar 

  26. Patil, N.S.: Coarse-grained models of nonlinear oscillators. PhD Thesis, The Pennsylvania State University, University Park, PA, USA (2017).

  27. Patil, N.S., Cusumano, J.P.: Empirical generating partitions of driven oscillators using optimized symbolic shadowing. Phys. Rev. E 98(3), 032211 (2018).

    Article  MathSciNet  Google Scholar 

  28. Petersen, K.E.: Ergodic theory. In: Cambridge Studies in Advanced Mathematics (No. 2). Cambridge University Press, Cambridge (1983)

  29. Politi, A.: Quantifying the dynamical complexity of chaotic time series. Phys. Rev. Lett. 118, 144101 (2017).

    Article  Google Scholar 

  30. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, New York (1992)

    MATH  Google Scholar 

  31. Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosso, O.A., Larrondo, H.A., Martin, M.T., Plastino, A., Fuentes, M.A.: Distinguishing noise from chaos. Phys. Rev. Lett. 99, 154102 (2007).

    Article  Google Scholar 

  33. Shalizi, C.R., Crutchfield, J.P.: Computational mechanics: pattern and prediction, structure and simplicity. J. Stat. Phys. 104, 817–879 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  34. Shalizi, C.R., Klinkner, K.L.: Blind construction of optimal nonlinear recursive predictors for discrete sequences. In: Chickering, M., Halpern, J.Y. (eds.) Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI 2004), pp. 504–511 (2004). arXiv:1408.2025

  35. Shalizi, C.R., Shalizi, K.L., Crutchfield, J.P.: An algorithm for pattern discovery in time series. SFI Working Paper 02-10-060, Santa Fe Institute (2002). arXiv:cs/0210025

  36. Solari, H.G., Gilmore, R.: Organization of periodic orbits in the driven Duffing oscillator. Phys. Rev. A 38, 1566–1572 (1988).

    Article  MathSciNet  Google Scholar 

  37. Sugihara, G., May, R.M.: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344(6268), 734–741 (1990).

    Article  Google Scholar 

  38. Tél, T., Lai, Y.C., Gruiz, M.: Noise-induced chaos: a consequence of long deterministic transients. Int. J. Bifurc. Chaos 18(02), 509–520 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  39. Zambella, D., Grassberger, P.: Complexity of forecasting in a class of simple models. Complex Syst. 2(3), 269–303 (1988)

    MathSciNet  MATH  Google Scholar 

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

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:

$$\begin{aligned} \displaystyle {{\varvec{y}}}_{r+1} = {{\varvec{y}}}_r + \sum _{i=1}^{6} \alpha _i h \, {{\varvec{f}}}(t_r+c_i h, H_i) +{{\varvec{g}}}\, I_{(1)}, \end{aligned}$$
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 \(i = 1,2,\ldots ,6\),

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

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:

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

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. 1920) of order 1.5, starting from a general scheme in [31]

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Patil, N.S., Cusumano, J.P. The high forecasting complexity of stochastically perturbed periodic orbits limits the ability to distinguish them from chaos. Nonlinear Dyn 102, 697–712 (2020).

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