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Codimension 3 heteroclinic bifurcations with orbit and inclination flips in reversible systems. (English) Zbl 1165.34358
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.

34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
34C14Symmetries, invariants (ODE)
37G15Bifurcations of limit cycles and periodic orbits
37C80Symmetries, equivariant dynamical systems
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