Generic behavior of master-stability functions in coupled nonlinear dynamical systems

Abstract

Master-stability functions (MSFs) are fundamental to the study of synchronization in complex dynamical systems. For example, for a coupled oscillator network, a necessary condition for synchronization to occur is that the MSF at the corresponding normalized coupling parameters be negative. To understand the typical behaviors of the MSF for various chaotic oscillators is key to predicting the collective dynamics of a network of these oscillators. We address this issue by examining, systematically, MSFs for known chaotic oscillators. Our computations and analysis indicate that it is generic for MSFs being negative in a finite interval of a normalized coupling parameter. A general scheme is proposed to classify the typical behaviors of MSFs into four categories. These results are verified by direct simulations of synchronous dynamics on networks of actual coupled oscillators.

IEEE Transactions on Control of Network Systems information for authors

“…The matrices DF and DH are the Jacobians of the vector field of the isolated system F(x) and coupling function H(x) respectively, evaluated on the synchronous solution s(t). For more details, please see for example [24], [25].…”

“…As an illustrative example, we will consider a network of coupled identical Rössler oscillators (as in [25]) with individual dynamics governed by: …”

“…"Connected flocking of multi-agent system based on distributed eigenvalue estimation". In: In networks of identical, coupled, nonlinear systems, as described by Equation (1), the predominant technique for determining the local stability of the synchronous solution is the Master Stability Function (MSF) approach [24], [25]. The MSF Ψ(α) plots the Largest Lyapunov Exponent (LLE) of the system governed by Equation (31): the dynamics of virtual displacement δy between two trajectories in the vicinity of the synchronous solution s(t), as a function of the coupling strength α between them.…”

“…The eigenvalues of the graph Laplacian are thus real, and can then be ordered as 0 = λ 1 < λ 2 ≤ · · · ≤ λ n . As shown in the previous literature [24], [25] and briefly summarized in Appendix A, local transversal stability of the synchronization manifold in a network of identical nonlinear systems can be studied using the Master Stability Function (MSF) approach (see Appendix A for further details). According to this approach, the range of coupling strengths that render the synchronous solution locally transversally stable can be maximized by maximizing the algebraic connectivity λ 2 or minimizing the Laplacian eigenratio r according to the specific shape of the MSF for the systems of interest.…”

Synchronization of nearly identical dynamical systems: Size instability

“…Pecora and Carroll have developed an elegant way, namely the master stability function (MSF) for analyzing the stability of complete synchronization for a network of identical dynamical systems [29]. The MSF allows one to study the stability of synchronization of different networks using a single function and has been used widely for a comparative study of synchronization of different networks of identical dynamical systems [30][31][32][33][34][35][36]. It is shown that the small world network enhances synchronizability of a network of coupled identical systems [31].…”