Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators

Systems of coupled oscillators may exhibit spontaneous dynamical formation of attractingsynchronized clusters with broken symmetry; this can be helpful in modelling various physicalprocesses. Analytical computation of the stability of synchronized cluster states is usually impos-sible for arbitrary nonlinear oscillators. In this paper we examine a particular class of stronglynonlinear oscillators that are analytically tractable. We examine the effect of isochronicity (aturning point in the dependence of period on energy) of periodic oscillators on clustered statesof globally coupled oscillator networks.We extend previous work on networks of weakly dissipative globally coupled nonlinear Hamil-tonian oscillators to give conditions for the existence and stability of certain clustered periodicstates under the assumption that dissipation and coupling are small and of similar order. This isverified by numerical simulations on an example system of oscillators that are weakly dissipativeperturbations of a planar Hamiltonian oscillator with a quartic potential.Finally we use the reduced phase-energy model derived from the weakly dissipative case tomotivate a new class of phase-energy models that can be usefully employed for understandingeffects such as clustering and torus breakup in more general coupled oscillator systems. We seethat the property of isochronicity usefully generalizes to such systems, and we examine someexamples of their attracting dynamics.