Master stability functions for coupled nearly identical dynamical systems

Abstract

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 significant roles played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems.

Synchronization in complex networks of phase oscillators: A survey

“…Among such phase oscillators networks, the often encountered and most thoroughly studied case is that of anti-symmetric coupling without higher-order harmonics, that is, the oscillator network (1) with sinusoidal coupling. Moreover, the coupled oscillator model (1) serves as the prototypical example for synchronization in complex networks (Strogatz, 2001;Boccaletti et al, 2006;Osipov et al, 2007;Suykens and Osipov, 2008;Arenas et al, 2008), and its linearization is the well-known consensus protocol studied in networked control, see the surveys and monographs (Olfati-Saber et al, 2007;Ren et al, 2007;Bullo et al, 2009;Garin and Schenato, 2010;Mesbahi and Egerstedt, 2010). Indeed, numerous control scientists explored the coupled oscillator model (1) as a nonlinear generalization of the consensus protocol (Jadbabaie et al, 2004;Moreau, 2005;Scardovi et al, 2007;Olfati-Saber, 2006;Lin et al, 2007;Chopra and Spong, 2009;Sarlette and Sepulchre, 2009;Sepulchre, 2011).…”

“…More general oscillator networks display the same phenomenology, but the threshold from incoherence to synchrony is generally unknown. Finally, we remark that for oscillator networks of dimension n ≥ 3, this loss of synchrony via a saddle-node bifurcation is only the starting point of a series of bifurcations occurring if the coupling is further decreased, see Tönjes, 2007;Popovych et al, 2005;Suykens and Osipov, 2008).…”

“…The network science, nonlinear dynamics, and physics communities coined the term complex for such non-trivial topologies to distinguish them from longrange (complete) and short-range (lattice-type) interaction topologies. The interest in such complex oscillator networks has been sparked by the seminal article (Jadbabaie et al, 2004) and the widespread scientific attention given to complex network studies (Strogatz, 2001;Boccaletti et al, 2006;Osipov et al, 2007;Arenas et al, 2008;Suykens and Osipov, 2008;Dorogovtsev et al, 2008), and consensus and its applications (Olfati-Saber et al, 2007;Ren et al, 2007;Bullo et al, 2009;Garin and Schenato, 2010;Mesbahi and Egerstedt, 2010).…”

“…Some of the proposed synchronization conditions for complex phase oscillator networks can be evaluated only numerically since they are state-dependent Kumagai, 1980, 1982) or arise from a non-trivial linearization process of full state space oscillator models. The latter procedure is adopted in the widely-studied Master Stability Function formalism, see (Pecora and Carroll, 1998;Boccaletti et al, 2006;Arenas et al, 2008) for the original reference and relevant surveys, see (Restrepo et al, 2004;Sun et al, 2009;Sorrentino and Porfiri, 2011) for its extension to quasi-identical oscillators, and see (Shafi et al, 2013;Russo and Di Bernardo, 2009) for related linearizationbased approaches from the control community.…”

“…In this case, there are various interesting transitions between wells of the potential landscape and only few analytic investigations. Also the sub-synchronous regime for K < K critical is vastly unexplored, and partial synchronization or clustering (similar to the infinite-dimensional model) De Smet and Aeyels, 2007), chimera states (Laing, 2009;Martens et al, 2013), or chaotic motion Tönjes, 2007;Popovych et al, 2005;Suykens and Osipov, 2008) can occur. Finally, the incremental stability results referenced in Subsection 4.1 appear to be a promising direction that still needs to be fully explored.…”