Detecting the dynamical instability of complex time series via partitioned entropy

A method is proposed to detect the dynamical instability of complex time series. We focus on how the partitioned entropy of an initially localized region of the attractor evolves in time and show that its growth rate corresponds to the first Lyapunov exponent. To avoid spurious detection of the dynamical instability, a criterion is further introduced to distinguish chaos from limit cycles or tori. Numerical experiments using prototypical models of chaotic systems demonstrate that the growth rate of the partitioned entropy indeed provides a good estimate of the first Lyapunov exponent. The method is also shown to be robust against observational noise and dynamical noise. Analysis of experimental data measured from a physical model of the vocal folds highlights the practical applicability of the present method to real-world data. Advantages of the present method over conventional methods are also discussed.

Figure 1

Illustration of an evolution of partitioned attractor. In two-dimensional space ($d=2$), an attractor exhibits a donut-like structure, which is partitioned into four cells ($M=4, i=1,2,3,4$). At time $t=0$, a set of initial conditions is covered by the first cell (shaded area on the left panel). As time $t$ passes, the set of initial states evolves into an ellipsoid (shaded area on the right panel), the diameter of which is expanded to $l_i\left(t\right)\approx \varepsilon e^{{\lambda }_1^{i}t}$. The number of cells required to cover the stretched ellipsoid can be approximated as $m_i\left(t\right)\approx e^{{\lambda }_1^{i}t}$.

Figure 2

Time evolution of system states on the reconstructed attractor in a two-dimensional space. The reconstructed attractor is partitioned into four cells (00, 01, 10, 11) by the threshold (dashed lines). The system states (solid circles) located within the cell 00 evolve in time, where the number of cells required to cover the spreading system states increases as time passes.

Figure 3

Time evolution of the partitioned entropy $h\left(t\right)$ in the case of (a) chaos and (b) limit cycle. The partitioned entropy is calculated from the $y$ variable of the Rössler equations (14) with $a=0.55, b=2$, and $c=4$ for chaos and $a=0.505, b=2$, and $c=4$ for limit cycle. The time series consist of $N=3\times {10}^5$ data points and the embedding dimension is set to $d=7$. The solid line represents a fitting function $\widehat{h}=min(\alpha t,\beta )$, whereas the dashed line represents the entropy of the whole attractor, $h_{\max }$.

Figure 4

Rössler system with a varying bifurcation parameter $a\in [0.45,0.55]$. (a) True value of the first Lyapunov exponent ${\lambda }_1$ calculated by the Shimada-Nagashima algorithm and estimate of the first Lyapunov exponent $\gamma$ obtained by the present method. (b) Ratio between the mean entropy and the attractor's entropy, $h_{\mathrm{mean}}/h_{\max }$. The dashed line indicates the threshold $r=0.8$, above which the system is considered chaotic.

Figure 5

Langford system with a varying bifurcation parameter $\varepsilon \in [0,0.25]$. (a) True value of the first Lyapunov exponent ${\lambda }_1$ calculated by the Shimada-Nagashima algorithm and estimate of the first Lyapunov exponent $\gamma$ obtained by the present method. (b) Ratio between the mean entropy and the attractor's entropy, $h_{\mathrm{mean}}/h_{\max }$. The dashed line indicates the threshold $r=0.8$, above which the system is considered chaotic.

Figure 6

Effects of noise added to the Rössler system. Observational noise is added to both [(a), (b)] chaotic and [(c), (d)] periodic time series, whereas [(e), (f)] dynamical noise is added to a periodic time series. In (a), (c), and (e), dependence of the estimated Lyapunov exponent $\gamma$ (solid line) and the true value ${\lambda }_1$ (dotted line) on the noise intensity is shown. In (b), (e), and (f), the ratio between the mean entropy and the attractor's entropy, $h_{\mathrm{mean}}/h_{\max }$, is drawn as a function of the noise intensity.

Figure 7

Application to experimental data from silicone model of the vocal folds. [(a), (c), (e)] Reconstructed attractors and [(b), (d), (f)] the time evolution of the partitioned entropy for #3P-a-2-1, #3P-a-3-2, and #3P-c-3-2, respectively.