Characterization of the chaos-hyperchaos transition based on return times

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.

Figure 1

Dependencies of ${\lambda }_1$ and ${\lambda }_2$ versus the parameter $c$ of the coupled Rössler oscillators (2). Circles indicate “true” values estimated with the method [1]. Squares are related to LEs estimated from the sequences of return times into the plane $x_1=0$. Diamonds denote LEs computed from two sequences of return times into the planes $x_1=0$ and $x_2=0$, when the restored attractor contains delayed coordinates related to the averaged instantaneous frequencies for both subsystems. Triangles are related to LEs estimated from a single sequence of return times into the plane $x_2+y_1=0$. The arrow marked by A shows the chaos-hyperchaos transition.

Figure 2

Computing two largest LEs depending from the secant plane introduced as $k_1{y}_2+k_2{x}_1=0$ or $k_1{y}_1+k_2{x}_2=0$, where $k_1=sin\alpha ,k_2=cos\alpha$. Dashed lines mark “true” values of LEs.

Figure 3

Dependencies of two largest LEs versus the parameter $c$ of the coupled Rössler oscillators (2) for the parameter ranges related to nearly close LEs. Circles indicate “true” values estimated with the method [1]. Triangles correspond to ${\lambda }_{1,2}$ computed from sequences of return times into the secant plane $x_2+y_1=0$ (2000 samples). Note that an underestimated ${\lambda }_2$ is computed from point processes that is caused by orientation errors.

Figure 4

Dependencies of two largest LEs versus the number of analyzed return times for the parameter $c=7.18$ of the coupled Rössler oscillators (2) and the secant plane $x_2+y_1=0$. Dotted lines indicate “true” values of ${\lambda }_{1,2}$ estimated with the method [1].