Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor

Departing from a system of two nonautonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a one-dimensional medium as an ensemble of such local elements introducing spatial coupling via diffusion. When length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero and the oscillations become inhomogeneous in space. However, at a point where the third Lyapunov exponent becomes positive, some bifurcation occur that results in violation of the hyperbolicity due to the emergence of one-dimensional intersections of contracting and expanding tangent subspaces along trajectories on the attractor. Further growth of the length results in the two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond the point of the hyperbolicity loss, the system demonstrates an extensive spatiotemporal chaos typical for extended chaotic systems: when the length of the system increases the Kaplan-Yorke dimension, the number of positive Lyapunov exponents and the upper estimate for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends to a limiting curve.

Figure 1

Growth rate $\sigma \left(k\right)$ of nonuniform perturbation to a homogeneous solution of the system (2). Solid curves are the graphs of Eq. (8): the upper one is $\left(\text{ln}2\right)/5-k^2$ and the lower one is $\sigma \left(0\right)-k^2$, where $\sigma \left(0\right)=0.00842$. Crosses, circles, and stars represent numerical data, computed at parameters that are shown in the legend.

Figure 2

Spatiotemporal dynamics of Eq. (2) at $L=8$. Gray levels indicate values of $\text{Re}a$. $A=3$, $T=5$, and $ϵ=0.05$.

Figure 3

Same as Fig. 3, but at $L=10$.

Figure 4

$L=500$.

Figure 5

Distributions of minimal angles ${\theta }_1$ between contracting and expanding tangent subspaces of the attractor of Eq. (2). $A=3$, $T=5$, and $ϵ=0.05$. The logarithmic scale is used along the ordinate axis. Dashed lines in the panels $L=17$ and $L=30$ are obtained via least-squares fit. There is one positive Lyapunov exponent at $L=8$, two at $L=10$ and $L=15$, three at $L=17$, five at $L=30$, and 11 at $L=60$. Observe the violation of hyperbolicity at $L>15$, i.e., when the third Lyapunov exponent becomes positive.

Figure 6

Distributions of two principal angles $\left({\theta }_1+{\theta }_2\right)/2$. Observe how curves approach the origin and touch it at $L=60$, which indicates the two-dimensional tangencies between contracting and expanding subspaces. The parameters are as in Fig. 5.

Figure 7

Distributions of ${\theta }_1$ and $\left({\theta }_1+{\theta }_2\right)/2$, panels (a) and (b), respectively, at $A=8$, $T=2$, and $ϵ=0.05$. There are two positive Lyapunov exponents at $L=10$, three at $L=11$, and 16 at $L=60$. Observe the violation of hyperbolicity at $L=11$ in the (a) and the emergence of two-dimensional tangencies at $L=60$ in (b).

Figure 8

Ten largest Lyapunov exponents of the system (2) against $L$. $A=3$, $T=5$, and $ϵ=0.05$. Vertical dashed lines mark $L_c\approx 8.44$ and $L_2\approx 15.9$ (the point where ${\lambda }_2=0$). Lower panel represents ${\lambda }_0$ in a large scale. Dashed approximating line, $4\times {10}^{-5}L+0.138$, is obtained via least-squares fit.

Figure 9

(a) Values of length $L_n$ where corresponding Lyapunov exponents ${\lambda }_n$ pass zero and (b) intervals between these points $\Delta L_n=L_n-L_{n-1}\left(\Delta L_1\equiv L_c\right)$. Solid lines on both panels correspond to the parameters $A=3$, $T=5$, and $ϵ=0.05$, and dashed lines represent parameters $A=8$, $T=2$, and $ϵ=0.05$.

Figure 10

Kaplan-Yorke dimension $D_{\text{KY}}$, upper estimate $h_{\mu }$ for the Kolmogorov-Sinai entropy, and its power law approximation ${\chi }_{\mu }$ (15) against $L$. (a) $A=3$, $T=5$, and $ϵ=0.05$. (b) $A=8$, $T=2$, and $ϵ=0.05$. Vertical dashed lines mark $L_c$ and $L_2$.

Figure 11

(a) Spectra of Lyapunov exponents and (b) scaled spectra at $A=3$, $T=5$, and $ϵ=0.05$. (c) Scaled spectra at $A=8$, $T=2$, and $ϵ=0.05$.