Symmetric Hopf bifurcation: twisted degree approach.

*(English)*Zbl 1192.34002
Battelli, Flaviano (ed.) et al., Handbook of differential equations: Ordinary differential equations. Vol. IV. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-53031-8/hbk). Handbook of Differential Equations, 1-131 (2008).

The subject of this survey article is the Andronov-Hopf bifurcation in dynamical systems admitting a certain group of symmetries. For this class of systems the authors have developed the method of twisted equivariant degree (TED) [Z. Balanov, W. Krawcewicz and H. Steinlein, Applied equivariant degree. Springfield, MO: American Institute of Mathematical Sciences (2006; Zbl 1123.47043)].

The introduction (Sec. 1) contains the history of the topological degree approach, Sec. 2 – the basic notions and definitions from representation theory (\(G\)-vector bundles and \(G\)-manifolds, Fredholm operators). Sec. 3 naturally splits into four parts: \(1^0\) main ideas underlying the notion of twisted degree (TDE), \(2^0\) its construction and basic properties, \(3^0\) practical computations, \(4^0\) infinite dimensional extentions. Sec. 4 gives the standard steps of the TDE treatment for an autonomous first order ODE system without symmetries. Here, according to the \(S^1\)-degree approach the occurrence of the Andronov-Hopf bifurcation in ODEs without symmetries is discussed together with the role of \(\Gamma\)-symmetries. The local bifurcation invariant (\(S^1\)-degree) is introduced with an effective formula for its computation. Sec. 5 is devoted to symmetric bifurcation for ODE systems with symmetries. Here, the concepts of dominating orbit types and isotypical crossing number are introduced, special Maple routines are developed for the TDE. Sec. 6 deals with a general symmetric Andronov-Hopf bifurcation problem in functional differential equations with application to the construction problem of symmetric branches of periodic solutions. In Sec. 7 the TDE method is extended to systems of functional parabolic differential equations. In Sec. 8 the results of Sec. 7 are illustrated by three concrete models: configurations of identical oscillators, symmetric configuration of transmission lines and symmetric system of the Hutchinson model in population dynamics.

Appendix A contains information about the dihedral group \(D_n\), icosahedral group \(A_5\) and their representations.

For the entire collection see [Zbl 1173.34001].

The introduction (Sec. 1) contains the history of the topological degree approach, Sec. 2 – the basic notions and definitions from representation theory (\(G\)-vector bundles and \(G\)-manifolds, Fredholm operators). Sec. 3 naturally splits into four parts: \(1^0\) main ideas underlying the notion of twisted degree (TDE), \(2^0\) its construction and basic properties, \(3^0\) practical computations, \(4^0\) infinite dimensional extentions. Sec. 4 gives the standard steps of the TDE treatment for an autonomous first order ODE system without symmetries. Here, according to the \(S^1\)-degree approach the occurrence of the Andronov-Hopf bifurcation in ODEs without symmetries is discussed together with the role of \(\Gamma\)-symmetries. The local bifurcation invariant (\(S^1\)-degree) is introduced with an effective formula for its computation. Sec. 5 is devoted to symmetric bifurcation for ODE systems with symmetries. Here, the concepts of dominating orbit types and isotypical crossing number are introduced, special Maple routines are developed for the TDE. Sec. 6 deals with a general symmetric Andronov-Hopf bifurcation problem in functional differential equations with application to the construction problem of symmetric branches of periodic solutions. In Sec. 7 the TDE method is extended to systems of functional parabolic differential equations. In Sec. 8 the results of Sec. 7 are illustrated by three concrete models: configurations of identical oscillators, symmetric configuration of transmission lines and symmetric system of the Hutchinson model in population dynamics.

Appendix A contains information about the dihedral group \(D_n\), icosahedral group \(A_5\) and their representations.

For the entire collection see [Zbl 1173.34001].

Reviewer: Oleg V. Makeev (Ulyanovsk)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |

34C23 | Bifurcation theory for ordinary differential equations |

34C14 | Symmetries, invariants of ordinary differential equations |

35B32 | Bifurcations in context of PDEs |

35R10 | Functional partial differential equations |

47N20 | Applications of operator theory to differential and integral equations |

34K18 | Bifurcation theory of functional-differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34K13 | Periodic solutions to functional-differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34K30 | Functional-differential equations in abstract spaces |