Using recurrences to characterize the hyperchaos-chaos transition

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

Figure 1

Time evolution (top) and recurrence plots (bottom) of the variable $x_n\left(2\right)$ for a coupled map lattice with $\beta =3$, $N=3$ and (a) $K=0.3$ and (b) $K=0.8$. We have used an embedding dimension $d=1$ and delay $\tau =1$, with a cutoff radius equal to 0.01 times the phase-space diameter.

Figure 2

Dependence of some recurrence quantifiers with the coupling strength for $N=3$ coupled map lattices with $\beta =3$: (a) recurrence rate; (b) determinism; (c) average diagonal length; (d) maximum diagonal length.

Figure 3

Dependence of some recurrence quantifiers with the coupling strength for $N=57$ coupled map lattices with $\beta =3$: (a) Recurrence rate; (b) determinism; (c) average diagonal length; (d) maximum diagonal length.

Figure 4

Dependence of the recurrence with the coupling strength for different values of the lattice size.

Figure 5

Bifurcation diagrams in the synchronization manifold for different lattice sizes: (a) $N=3$; (b) $N=19$; (c) $N=57$. The sudden increase of the density of points coincides with the hyperchaos-chaos transition.

Figure 6

Dependence of some recurrence quantifiers with the coupling strength for $N=3$ coupled map lattices with $\beta =3$ and $K$ is randomly changed: (a) Recurrence rate; (b) determinism; (c) average diagonal length; (d) maximum diagonal length.

Figure 7

Dependence of some recurrence quantifiers with the coupling strength for two unidirectionally coupled Chua-Matsumoto circuits using the same parameters as in Ref. 18 recurrence rate; (b) determinism; (c) average diagonal length; (d) maximum diagonal length. The dashed line indicates the hyperchaos-chaos transition.