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A subshift of finite type in the Takens-Bogdanov bifurcation with \(D_3\) symmetry. (English) Zbl 0929.34040

Summary: The author studies the versal unfolding of a vector field of codimension two, that has an algebraically double eigenvalue \(0\) in the linearization of the origin and is equivariant under a representation of the symmetry group \(D_3\). A subshift of finite type is encountered near a clover of homoclinic orbits. The subshift encodes the itinerary along the three different homoclinic orbits. In this subshift all those symbol sequences are realized for which consecutive symbols are different. In the parameter space it is located a transcritical, three different Hopf and two global (homoclinic) bifurcations.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C28 Complex behavior and chaotic systems of ordinary differential equations
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