Master stability functions for coupled nearly identical dynamical systems

We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear-stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations. Our analysis underlines the significant role played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems. Copyright c © EPLA, 2009 Introduction. – Synchronization in its various forms has been a highly popular and exciting developing topic in the recent literature on chaotic oscillators [1,2]. Applications have ranged widely from biology [3,4] to mathematical epidemiology [5], and from modelling animal gaits [6] to engineering of communications devices [1,7], including many developments in complex networks (see e.g., refs. [8–12] and a review [13]). However, the preponderance of the work has focused on identical synchronization since it is in this situation whereby a complete analysis can be carried forward by the master stability formalism developed in the seminal work in ref. [14]. While other forms of synchronization are discussed in the literature, of particular interest here is nearly-synchronous state behavior of the systems that are slightly detuned from identical synchronization, which may or may not be associated with an invariant manifold [15] normally required to describe generalized synchrony [16–18]. In this letter we consider a coupled dynamical system consisting of N units coupled through some underlying network. The equations of motion read ẇi = f(wi, μi)− g N ∑ j=1 LijH(wj), i= 1, 2, . , N, (1) where f :"→" is the parameterized dynamics of an isolated unit; wi ∈" is the dynamical variable for (a)E-mail: sunj@clarkson.edu the i-th unit; μi ∈" is the corresponding parameter; L∈" is the graph Laplacian (unnormalized)1; H : "→" is a uniform coupling function; and g ∈" is the uniform coupling strength (usually > 0 for diffusive coupling). Note that we can represent the whole system conveniently by using the Kronecker product representation: ẇ= f(w,μ)− g ·L⊗H(w), (2) where w= [w 1 , w T 2 , . , w T N ] T is a column vector of all the dynamic variables, and likewise for μ and f ; and ⊗ is the usual Kronecker product (or direct product) [21]. System (1) has been studied mostly in the case in which the parameter μi is the same for each individual oscillator, often resulting in identical synchronization where max i,j ‖wi(t)−wj(t) ‖→ 0 as t→∞. The stability of such states can be analyzed by master stability functions (MSF) [14]. However, a noiseless system with exactly the same parameters is impossible in practice. It is known that parameter mismatch among the individual oscillators can cause bursts due to the instability of typical periodic orbits embedded in the synchronized chaotic attractor [9]; even 1We only deal with graph Laplacians that are diagonalizable for the sake of clarity. Non-diagonalizable Laplacians can be treated by techniques proposed in [19,20].