Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems

In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two “parameters of order,” i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.

Figure 1

Star-type configuration of coupling.

Figure 2

Chain-type configuration of coupling.

Figure 3

The comparison of the synchronization threshold (ratio $d_s∕{\lambda }_1$ versus the number of oscillators in chain) calculated from DSSM eigenvalues analysis and obtained from numerical investigations of chain synchronization; $d_s$, synchronization value of the coupling coefficient. (a) Duffing oscillators, (b) Lorenz systems, (c) Rössler systems.

Figure 4

The comparison of the synchronization ranges of coupling coefficient $d$ in the chain of diffusively coupled logistic maps calculated from DSSM eigenvalues analysis and obtained from numerical simulations; $d_u$ and $d_l$, upper and lower ends of the synchronization range.

Figure 5

Four oscillators with irregular configuration of coupling.

Figure 6

Bifurcation diagrams (the sum of average trajectories separation vs coupling coefficient) representing the comparison of the synchronization ranges in the ensembles of dynamical systems [(a) set of Rössler systems, (b) set of logistic maps] with the scheme of connections shown in Fig. 5. The ranges obtained analytically according to Eqs. (45, 46) are marked and described in black.

Figure 7

Bifurcation diagram of average trajectories separations vs coupling coefficient (a) and corresponding eigenvalues of DSSM [Eq. (48)] (b) for ten Duffing oscillators with random structure of coupling [Eq. (47)].