Experimental and Numerical Evidence for Riddled Basins in Coupled Chaotic Systems.

Recent work by Alexander et al. [1] has shown that theoretically there can exist chaotic dynamical systems having chaotic attractors that have their basins of attraction literally riddled with points from another attractor's basin. Basin riddling frequently occurs when the chaotic attractor exists in an invariant subspace (e.g. , a symmetry plane) and the second attractor is not contained within this subspace. The riddling is extreme in that the points of the other basin are dense in the chaotic attractor's basin (existing arbitrarily close to the chaotic attractor) and have a nonzero measure. These properties imply that any initial condition in the neighborhood of a point going to the chaotic attractor has a nonzero probability of going to the other attractor. Consequently, predictability of the final state is much more difficult than in systems with "ordinary" fractal basin boundaries. Several theoretical and numerical works have appeared that support these findings [2—8]. Lai and Winslow [9,10] have also found that some systems can exhibit riddling in parameter space. An important issue is whether these effects can be seen in real physical systems. The purpose of this Letter is to address this issue. Ott and Sommerer [4] have conjectured that one place to look for riddled basins is in reaction-diffusion systems of the type u = V2u + f(u) or their discrete counterparts, diffusively coupled oscillator systems. The latter systems can exhibit synchronous chaotic behavior, and are the systems we consider here. In the framework of diffusively coupled oscillator systems, the synchronization manifold [11],defined simply as the space where all oscillators execute identical dynamics, plays the role of the invariant subspace where the chaotic attractor exists. Motion on it is governed by u = f(u) (since V2u = 0 in the synchronous case). Stability of the synchronous chaotic attractor is determined by the transverse Lyapunov exponents [1—7,11]. When the largest of these exponents approaches zero from below there will be points in the synchronization manifold that are dense in the attractor (but of measure zero) and that are repelling from the attractor in directions transverse to the manifold. If there is another (nonsynchronous) attractor in the system, points in the neighborhood of the synchronization manifold can be repelled onto the other attractor. Hence, the second attractor has basin points arbitrarily close to the chaotic attractor —the basin of the synchronous attractor is then said to be riddled [12]. Basin riddling is of possible importance in systems where synchronous chaotic behavior has been seen or is expected. There are numerous examples, making a complete list impossible. Representative works are synchronous chaotic lasers [13,14], synchronized chaotic circuits or communications systems [15—17], synchronized, but chaotic biological systems [18,19], and studies using standard chaotic systems coupled in some fashion [20,21]. Recently, Ashwin et al. [7] have begun experiments with two coupled chaotic circuits that have shown that near the synchronization threshold there is a "bubbling" behavior of the circuit's motion, suggesting repelling points on the attractor in directions away from the synchronous state. The authors make a case for basin riddling in their system through empirically determined normal Lyapunov exponents obtained from a map constructed from the experimental data. It appears, however, that direct experimental observation of riddled basins and the scaling behavior associated with them [3,5] has not been carried out. We report here on direct observations of riddled basins in experiments and associated numerical work with a system of synchronized chaotic circuits [22]. The structure of the numerical riddling matches well with the experimental basin picture. Furthermore, both the experimental and numerical results display a scaling of the measure of the basin riddling in accord with existing theory [3,5]. In previous work [11] we showed that diffusively coupled systems with periodic boundary conditions are members of a class of coupled systems with so-called shift symmetry. For W oscillators this symmetry allows one to decompose the behavior near the synchronous state into N spatial Fourier modes from 0 to N/2 with 1 through N/2