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Effects of network structure on the synchronization of Hamiltonian systems

Yogesh Virkar

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Date and time:Ìý

Thursday, November 20, 2014 - 2:00pm

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ECCR 257

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The Hamiltonian Mean Field (HMF) model is a prototype used to model systems with long-range interactions such as gravitational systems, free-electron lasers, etc. The study of such systems has received a lot of attention due to their interesting dynamical properties such as phase transitions, persistent meta-equilibrium states, violent relaxation, etc.Ìý The HMF models a system of Hamiltonian rotors with a moment of inertia and coupled by interactions that promote their alignment.Ìý

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A significant amount of research on the HMF has been devoted to study the all-to-all case (each rotor connected to every other rotor with equal coupling strength). However, real-world systems such as the gravitational systems are better described by generalized network structures. Hence we consider the general case where the interactions between the rotors are governed by a complex network structure. We study analytically and numerically the HMF model on different network structures including the Erd\"os-Renyi and scale-free networks. We define a set of local order parameters and find a set of self-consistent nonlinear equations that they satisfy. From these equations we obtain conditions for the onset of synchronization in terms of the network coupling matrix. Our findings reveal that the network structure plays a crucial role in determining the onset of synchronization. The path to synchrony is gradual for heterogenous networks and sharper for homogenous networks. However, we find that for a fixed setting of the HMF model parameters, the maximum possible synchrony is fairly independent of the network structure.Ìý