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- Title
Coupled Oscillatory Systems with Symmetry and Application to van der Pol Oscillators.
- Authors
Murza, Adrian C.; Yu, Pei
- Abstract
In this paper, we study the dynamics of autonomous ODE systems with symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic -equivariant systems. Then, we analyze the action of on and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs of spatiotemporal or spatial symmetries, and prove their existence by using the Theorem due to Hopf bifurcation and the symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in -equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with 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.
- Subjects
VAN der Pol oscillators (Physics); MATHEMATICAL symmetry; ORDINARY differential equations; HOPF bifurcations; PROOF theory; PARAMETRONS
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2016, Vol 26, Issue 8, p-1
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127416501418