A Subshift of Finite Type in the Takens-Bogdanov Bifurcation with $D_3$ Symmetry

We study the versal unfolding of a vector field of codimension two, that has an algebraically double eigenvalue $0$ in the linearisation 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 we also locate a transcritical, three different Hopf and two global (homoclinic) bifurcations.

1991 Mathematics Subject Classification: {34C28, 58F14}

Keywords and Phrases: Takens-Bogdanov, subshift of finite type, symmetry, homoclinic orbit

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