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Coupled Oscillatory Systems with 픻4 Symmetry and Application to van der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with 𝔻4 symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic 𝔻4-equivariant systems. Then, we analyze the action of 𝔻4 on ℂ2 and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs (H,K) of spatiotemporal or spatial symmetries, and prove their existence by using the HmodK Theorem due to Hopf bifurcation and the 𝔻4 symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in 𝔻4-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with 𝔻4 symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.