Kokubu, Hiroshi; Komuro, Motomasa; Oka, Hiroe Multiple homoclinic bifurcations from orbit-flip. I: Successive homoclinic doublings. (English) Zbl 0884.34047 Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, No. 5, 833-850 (1996). The authors study families of piecewise affine vector fields in \(\mathbb{R}^3\) 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. Reviewer: Ale Jan Homburg (Berlin) Cited in 13 Documents MSC: 34C23 Bifurcation theory for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations Keywords:homoclinic orbit; cascades of homoclinic-doubling bifurcations; piecewise affine vector fields Software:UBASIC PDFBibTeX XMLCite \textit{H. Kokubu} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, No. 5, 833--850 (1996; Zbl 0884.34047) Full Text: DOI