Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

Figure 1

(a) Time realization of quadratic map (3) for $\mathrm{\Omega }=0.1$, $\varepsilon =0.008$, $D=0.0001$; (b) probability density $\rho \left(x\right)$ (histogram 1) obtained by time realization $x_n$ and its approximation by regularity (6) at $A=1.07\times {10}^{-13}$, $\mathrm{\Omega }=0.111124$, $\varepsilon =0.0087$, $D=0.0001$ (solid line), and analogous probability density $\rho \left(x\right)$ (histogram 2) obtained by time realization $x_n$ for $\mathrm{\Omega }=0.1$, $\varepsilon =0.0$, $D=0.0001.$

Figure 2

Dependencies of (a) Lyapunov exponent ${\mathrm{\Lambda }}_0$ and (b) the relative error $\delta$ for the criticality parameter $\varepsilon$ for the circle map (1) with $\mathrm{\Omega }=0.1$ in the presence of noise. In panel (a) data obtained by means of the proposed method are marked by filled circles; the results of application of a standard algorithm are shown by the solid line.

Figure 3

(a) Time dependence of the phase difference $\mathrm{\Delta }\phi \left(t\right)$ of the nonautonomous Van der Pole generator (10) for $D=1$, $\epsilon =0.043$; (b) probability density $\rho \left(\mathrm{\Delta }\phi \right)$ (histogram) obtained by the phase difference $\mathrm{\Delta }\phi \left(t\right)$ and its approximation by regularity (6) at $A=1.47\times {10}^{-82}$, $\mathrm{\Omega }=0.00486$, $\varepsilon =0.0029$, $D=0.0005$ (solid line).

Figure 4

Dependencies of (a) conditional Lyapunov exponent ${\mathrm{\Lambda }}_0$ corresponding to the enslaved phase dynamics of nonautonomous Van der Pole generator (10) in the presence of noise ($D=1$) and (b) the relative error $\delta$ on the external signal amplitude $\epsilon$. In panel (a) data obtained by means of the proposed method are marked by filled circles; the results of application of Benettin algorithm are shown by the solid line. The boundary of the synchronous regime in panel (b) is shown by the arrow.

Figure 5

(a) Time dependence of the phase difference $\mathrm{\Delta }\phi \left(t\right)$ of Rössler oscillators (11) for $\sigma =0.07$; (b) probability density $\rho \left(\mathrm{\Delta }\phi \right)$ (histogram) obtained by the phase difference $\mathrm{\Delta }\phi \left(t\right)$ and its approximation by regularity (6) at $A=4.97\times {10}^{-54}$, $\mathrm{\Omega }=0.0048$, $\varepsilon =0.0617$, $D=0.0024$ (solid line).

Figure 6

Dependencies of (a) conditional Lyapunov exponent ${\mathrm{\Lambda }}_0$ corresponding to the enslaved phase dynamics of Rössler oscillators (11) and (b) the relative error $\delta$ on the coupling parameter $\sigma$. In panel (a) data obtained by means of the proposed method are marked by points, the results of application of Benettin algorithm with Gram-Schmidt orthogonalization procedure are shown by solid line. The boundary of the phase synchronization regime ${\sigma }_{PS}$ in panel (b) is shown by the arrow.