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Multiple homoclinic bifurcations from orbit-flip. I: Successive homoclinic doublings. (English) Zbl 0884.34047
The authors study families of piecewise affine vector fields in 3 that unfold an orbit-flip homoclinic orbit. The main interest of the paper lies in the occurence of cascades of homoclinic-doubling bifurcations. The authors introduce and study a model one-dimensional map which seems to describe the main aspects of the dynamics, if the strong stable eigenvalue at the singularity in the origin is large in modulus. This study and numerical evidence make plausible that cascades of homoclinic-doubling bifurcations exist in the family of piecewise affine vector fields. Possible universal scaling behavior is discussed on the one dimensional map.
MSC:
34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
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