Generalized synchronization of chaos in autonomous systems

We extend the concept of generalized synchronization of chaos, a phenomenon that occurs in driven dynamical systems, to the context of autonomous spatiotemporal systems. It means a situation where the chaotic state variables in an autonomous system can be synchronized to each other, but not to a coupling function defined from them. The form of the coupling function is not crucial; it may not depend on all the state variables. Nor does it need to be active for all times for achieving generalized synchronization. The procedure is based on an analogy between a response map subject to an external drive acting with a probability $p$ and an autonomous system of coupled maps where a global interaction between the maps takes place with this same probability. It is shown that, under some circumstances, the conditions for stability of generalized synchronized states are equivalent in both types of systems. Our results reveal the existence of similar minimal conditions for the emergence of generalized synchronization of chaos in driven and in autonomous spatiotemporal systems.

Figure 1

Lyapunov exponents ${\Lambda }_y$ and ${\Lambda }_x$ for the driven system, Eq. (1), as a function of $p$, for $g\left(y_t\right)=0.5+\ln \mid y_t\mid$ and $f\left(x_t\right)=-0.7+\ln \mid x_t\mid$, and $ϵ=0.7$.

Figure 2

Orbits of the driven map, Eq. (1), for $g\left(y_t\right)=0.5+\ln \mid y_t\mid$ and $f\left(x_t\right)=-0.7+\ln \mid x_t\mid$, $ϵ=0.7$. (a) $p=0.3$ (unsynchronized). (b) $p=0.9$ (generalized synchronization). The inset is a magnification of the marked square.

Figure 3

Boundaries ${\Lambda }_x=0$ [Eq. (6)] on the plane $(p,ϵ)$ for complete synchronization in the driven map, Eq. (1), with $f\left(x_t\right)=-0.7+\ln \mid x_t\mid$. Dashed line: $g\left(y_t\right)=\{{\overline{x}}_1=-0.855762\}$. Solid line: $g\left(y_t\right)=-0.7+\ln \mid y_t\mid$ [this boundary also corresponds to complete synchronization in the autonomous system Eqs. (7, 9)]. Dotted line: $g\left(y_t\right)=\{{\overline{x}}_1=0.18049,{\overline{x}}_2=-2.41208\}$.

Figure 4

Region for generalized synchronized states $x_t\ne C$ (satisfying ${\Lambda }_x<0$) for the system, Eq. (1), subject to a constant drive $g\left(y_t\right)=C$, on the plane $(C,ϵ)$. Horizontal lines indicate the values $C=2$, $C=0$, and $C=\{{\overline{x}}_1=-0.855762\}$. The synchronization region for $C⩾0$ also comprises the generalized synchronized states $(⟨\sigma ⟩<{10}^{-7})$ of the autonomous system, Eq. (7), with $N={10}^4$, having a coupling function given by Eq. (13). In both cases, $f\left(x\right)=-0.7+\ln \mid x\mid$ and $p=0.5$.

Figure 5

Left vertical axis: bifurcation diagrams of $S_t\left(N\right)$ (black dots) and $H$ (gray dots) vs $ϵ$ for the autonomous system given by Eqs. (7, 13) with $f\left(x_t^i\right)=-0.7+\ln \mid x_t^i\mid$. Right vertical axis: $⟨\sigma ⟩$ vs $ϵ$ (solid line). Fixed parameters: $p=0.5$, $N={10}^4$.

Figure 6

Region of generalized synchronization on the plane $(p,ϵ)$ for (i) the single driven map, Eq. (1), with $g=0$ $({\Lambda }_x<0)$ and (ii) the autonomous system, Eq. (7), with $N={10}^4$ and $H$ given by Eq. (15) for any value of $q$ $(⟨\sigma ⟩<{10}^{-7})$. In both cases, $b=-0.7$.

Figure 7

Region for generalized synchronization on the plane $(ϵ,\gamma )$ for both the autonomous network, Eq. (16), with $H$ given by Eq. (13) with $q=200$, $r=500$, and the driven network, Eq. (17), with constant $g=0.4$. For both systems, $f\left(x\right)=4x(1-x)$, $N={10}^3$, and $p=0.5$.