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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.

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