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Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay. (English) Zbl 1223.34102

Subject of the paper is the system of three coupled van der Pol oscillators

x ¨ k +x k -ε(1-x k 2 )x ˙ k =k[x k+1 (t-τ)-2x k (t-τ)+x k-1 (t-τ)],k=0,1,2,

where the indices are considered modulo 3. Due to the symmetric coupling structure, this system is equivariant with respect to the D 3 group action. Using symmetric bifurcations theory, the authors investigate the Hopf bifurcations of the equilibrium x 1 =x 2 =x 3 =0. As a result of these bifurcations, the following periodic rotating waves appear: mirror-reflecting waves, standing waves, and discrete waves. The paper discusses the appearance and stability of these waves.

MSC:
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34C15Nonlinear oscillations, coupled oscillators (ODE)
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