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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 D. M. Zhu [Acta Math. Sin., Engl. Ser. 14, 341–352 (1998; Zbl 0932.37032)], D. M. Zhu and 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:
 34C37 Homoclinic and heteroclinic solutions of ODE 34C23 Bifurcation (ODE) 34C14 Symmetries, invariants (ODE) 37G15 Bifurcations of limit cycles and periodic orbits 37C80 Symmetries, equivariant dynamical systems