Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3. (English) Zbl 1165.34026
This paper deals with heteroclinic cycles connecting hyperbolic equilibria in \(\mathbb R^3\) having different indices (dimension of the unstable manifold). Consequently one of the heteroclinic connections lies in the intersection of the corresponding one-dimensional manifolds of the equilibria, and the second heteroclinic orbit (\(\Gamma_1\)) is in the intersection of the two-dimensional stable and unstable manifolds of the equilibria. It is assumed that the two-dimensional manifolds coincide along \(\Gamma_1\), other degeneracies are excluded. The authors claim that then the entire cycle is of codimension three.
Under the assumption that the eigenvalues of the equilibria are real the authors study bifurcating 1-homoclinic, 1-periodic and 2-periodic orbits.
The analysis is based on a first return map, which again is constructed using techniques from [D. Zhu and Z. Xia, Sci. China, Ser. A 41, No. 8, 837–848 (1998; Zbl 0993.34040)].
Under the assumption that the eigenvalues of the equilibria are real the authors study bifurcating 1-homoclinic, 1-periodic and 2-periodic orbits.
The analysis is based on a first return map, which again is constructed using techniques from [D. Zhu and Z. Xia, Sci. China, Ser. A 41, No. 8, 837–848 (1998; Zbl 0993.34040)].
Reviewer: Jürgen Knobloch (Ilmenau)
MSC:
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 37G25 | Bifurcations connected with nontransversal intersection in dynamical systems |
| 34C23 | Bifurcation theory for ordinary differential equations |
Citations:
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