Bifurcation analysis of a heterogeneous mean-field oscillator game model

This paper studies the phase transition in a heterogeneous mean-field oscillator game model using methods from bifurcation theory. In our earlier paper [1], we had obtained a coupled PDE model using mean-field approximation and described linear analysis of the PDEs that suggested possibility of a Hamiltonian Hopf bifurcation. In this paper, we simplify the analysis somewhat by relating the solutions of the PDE model to the solutions of a certain nonlinear eigenvalue problem. Both analysis and computations are much easier for the nonlinear eigenvalue problem. Apart from the bifurcation analysis that shows existence of a phase transition, we also describe a Lyapunov-Schmidt perturbation method to obtain asymptotic formulae for the small amplitude bifurcated solutions. For comparison, we also depict numerical solutions that are obtained using the continuation software AUTO.