Synchronization of nearly identical dynamical systems: Size instability

Abstract

We study the generalized synchronization and its stability using master stability function (MSF), in a network of coupled nearly identical dynamical systems. We extend the MSF approach for the case of degenerate eigenvalues of the coupling matrix. Using the MSF we study the size-instability in star and ring networks for coupled nearly identical dynamical systems. In the star network of coupled Rössler systems we show that the critical size beyond which synchronization is unstable, can be increased by having a larger frequency for the central node of the star. For the ring network we show that the critical size is not significantly affected by parameter variations. The results are verified by explicit numerical calculations.

Frustration induced transient chaos, fractal and riddled basins in coupled limit cycle oscillators

“…The stable boundary for synchronized region obtained through master stability analysis [55,56,57]. In the synchronized manifold, x i = x, and y i = y, ∀i, then the variational equation of system (1) can be written as,η 1j = η 2j + ǫ 1 λ j η 1j , j = 1, 2, 3, ...N,…”

From approximate synchronization to identical synchronization in coupled systems

“…On the other hand, depending on the methodologies employed, some of those investigations conclude local dynamics, whereas others established global dynamics. In particular, the works which analyze the stability of synchronous manifold or near-synchronous manifold via linearization and the ones involving master stability function conclude local dynamics, see also [1,2,49,51]. On the other hand, Lyapunov function theory and ideas developed from this theory are typical tools for concluding global synchronization.…”

“…. , x N (t)) is the corresponding solution of system (2). System (2) reduces to ordinary differential equations when τ M = 0.…”

“…While our approach can treat system (2), the formulation is very cumbersome. So in this paper, we illustrate our approach by considering system (2)…”

Synchronization-optimized networks for coupled nearly identical oscillators and their structural analysis

“…Thus, to better understand synchronization processes of natural systems it is important to construct a master stability function for generalized synchronization. Motivated by this problem, we have extended the formalism of the master stability function to the generalized synchronization of coupled nearly identical oscillators [8][9][10].…”