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Zbl 1165.34358
Xu, Yancong; Zhu, Deming; Geng, Fengjie
Codimension 3 heteroclinic bifurcations with orbit and inclination flips in reversible systems.
(English)
[J] Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, No. 12, 3689-3701 (2008). ISSN 0218-1274

Summary: Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in {\it D. M. Zhu} [Acta Math. Sin., Engl. Ser. 14, 341--352 (1998; Zbl 0932.37032)], {\it D. M. Zhu} and {\it Z. H. Xia} [Sci. China, Ser. A 41, 837--848 (1998; Zbl 0993.34040)]. The existence and coexistence of $R$-symmetric homoclinic orbit and $R$-symmetric heteroclinic orbit, $R$-symmetric homoclinic orbit and $R$-symmetric periodic orbit are obtained. The double $R$-symmetric homoclinic bifurcation is found, and the continuum of $R$-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.
MSC 2000:
*34C37 Homoclinic and heteroclinic solutions of ODE
34C23 Bifurcation (periodic solutions)
34C14 Symmetries, invariants
37G15 Bifurcations of limit cycles and periodic orbits
37C80 Symmetries, equivariant dynamical systems

Keywords: heteroclinic bifurcation; orbit flip; inclination flip; reversible system

Citations: Zbl 0932.37032; Zbl 0993.34040

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