Asymptotic stability of pseudo-simple heteroclinic cycles in \(\mathbb R^4\). (English) Zbl 1386.34083
The authors consider vector fields defined in \(\mathbb R^4\) which commute with the action of a group \(\Gamma\) of isometries of \(\mathbb R^4\):
\[
\dot{x}=f (x),\text{ where }(\gamma x)=\gamma f(x)\text{ for all }\gamma \in \Gamma,\, \Gamma \subset O(4)\text{ is finite}
\]
and \(f\) is a smooth map in \(\mathbb R^4\). These kind of vector fields can possess robust heteroclinic cycles. The properties of these objects in the case of a vector field of \(\mathbb R^3\) has been already studied and a complete classification has been done. The dynamics in \(\mathbb R^4\) are more intricate. In an article [M. Krupa and I. Melbourne, Ergodic Theory Dyn. Syst. 15, No. 1, 121–147 (1995; Zbl 0818.58025)] a definition of simple robust heteroclinic cycle is provided. Let \(\Delta_j\) denote the isotropy subgroup of an equilibrium \(\xi_j\). The heteroclinic cycle is simple if
The aim of the work is to investigate the dynamical properties of pseudo-simple heteroclinic cycles in \(\mathbb R^4\), such as stability and asymptotic stability. Some particular examples are analyzed.
- (i)
- the fixed-point subspace of \(\Delta_j\), denoted \(\mathrm{Fix}(\Delta_j)\), is an axis;
- (ii)
- for each \(j\) , \(\xi_j\) is a saddle and \(\xi_{j+1}\) is a sink in an invariant plane \(\mathrm{Fix}(\Sigma_j)\) (hence \(\Sigma_j \subset \Delta_j \cap \Delta_{j+1}\));
- (iii)
- the isotypic decomposition for the action of \(\Delta_j\) in the tangent space at \(\xi_j\) only contains one-dimensional components.
The aim of the work is to investigate the dynamical properties of pseudo-simple heteroclinic cycles in \(\mathbb R^4\), such as stability and asymptotic stability. Some particular examples are analyzed.
Reviewer: Maite Grau (Lleida)
MSC:
34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
34C14 | Symmetries, invariants of ordinary differential equations |
37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
34D20 | Stability of solutions to ordinary differential equations |
34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
robust heteroclinic cycles; equivariant dynamical systems; asymptotic stability; periodic orbits; bifurcationsCitations:
Zbl 0818.58025References:
[1] | Chossat, P.: The bifurcation of heteroclinic cycles in systems of hydrodynamical type. J. Contin. Discrete Impuls. Syst. 8a(4), 575-590 (2001) · Zbl 1030.37057 |
[2] | Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific Publishing Company, Singapore (2000) · Zbl 0968.37001 · doi:10.1142/4062 |
[3] | Du Val, P.: Homographies, Quaternions and Rotations. OUP, Oxford (1964) · Zbl 0128.15403 |
[4] | Krupa, M.: Robust heteroclinic cycles. J. Nonlinear Sci. 7, 129-176 (1997) · Zbl 0879.58054 · doi:10.1007/BF02677976 |
[5] | Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergod. Theory Dyn. Syst. 15, 121-148 (1995a) · Zbl 0818.58025 · doi:10.1017/S0143385700008270 |
[6] | Krupa, M.; Melbourne, I.; Langford, WF (ed.); Nagata, W. (ed.), Nonasymptotically stable attractors in O(2) mode interaction, No. 4, 219-232 (1995), Providence · Zbl 0833.58025 · doi:10.1090/fic/004/11 |
[7] | Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry. II. Proc. R. Soc. Edinb. 134A, 1177-1197 (2004) · Zbl 1073.37025 · doi:10.1017/S0308210500003693 |
[8] | Podvigina, O.M.:. Stability and bifurcations of heteroclinic cycles of type Z. Nonlinearity 25, 1887-1917 (2012). arXiv:1108.4204 [nlin.CD] · Zbl 1259.37015 |
[9] | Podvigina, O.M.: Classification and stability of simple homocliniccycles in \[{\mathbb{R}}^5\] R5. Nonlinearity 26, 1501-1528 (2013). arXiv:1207.6609 [nlin.CD] · Zbl 1278.34051 |
[10] | Podvigina, O.M., Chossat, P.: Simple heteroclinic cycles in \[\mathbb{R}^4\] R4. Nonlinearity 28, 901-926 (2015). arXiv:1310.0298 [nlin.CD] · Zbl 1351.34050 |
[11] | Podvigina, O.M., Ashwin, P.: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity 24, 887-929 (2011). arXiv:1008.3063 [nlin.CD] · Zbl 1219.37021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.