Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach
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Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach Zahra Aminzare1 · Biswadip Dey2 · Elizabeth N. Davison2 · Naomi Ehrich Leonard2 Received: 5 July 2017 / Accepted: 21 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Finding the conditions that foster synchronization in networked nonlinear systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with FitzHugh–Nagumo dynamics, we show that our new sufficient condition is tighter than those found in previous analyses that used smooth or nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network
Communicated by Danielle S. Bassett.
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Zahra Aminzare [email protected] Biswadip Dey [email protected] Elizabeth N. Davison [email protected] Naomi Ehrich Leonard [email protected]
1
The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
2
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540, USA
123
J Nonlinear Sci
configuration is a significant step toward facilitating understanding and control of complex networked systems. Keywords Cluster synchronization · Contraction theory for stability · Diffusively coupled nonlinear networks · Neuronal oscillators
1 Introduction Synchronization has been observed and studied in diverse fields. Its presence has been characterized in symmetric networks of identical mechanical systems or identical biological systems, as well as those with differing types of individual components and nonuniform coupling (Pikovsky et al. 2003). The role of synchronization has been studied in a multitude of both natural and engineered settings including collective motion (Sepulchre et al. 2008), power-grid networks (Motter et al. 2013), robotics (Nair and Leonard 2008), sensor networks (Sivrikaya and Yener 2004), circadian rhythms (Winfree 1967), bioluminescence in fireflies (Smith 1935), pacemaker cells in the heart (Mirollo and Strogatz 1990), neuronal ensembles (Chow and Kopell 2000), and numerous others. In the human brain, synchronization at the neuronal or regional level can be beneficial, allowing for production of a vast range of behaviors (Dumas et al. 2010; MacLeod and Laurent 1996), or detrimental, causing disorders suc
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