Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation.

*(English)*Zbl 0952.37009The authors study a two-parameter unfolding of a reversible four-dimensional differential system which undergoes a so-called reversible Hopf bifurcation (the authors call it the reversible Hamiltonian-Hopf bifurcation), i.e., when the system possesses an equilibrium with double nonsemisimple imaginary eigenvalues. This requires a special type of involution action with a two-dimensional fixed plane. Degeneracy here means that a coefficient, which is responsible for the type of bifurcation, in the third order normal form at the equilibrium vanishes. The theoretical analysis is given for the third order normal from for the unfolding. This normal form is integrable (as well as the normal form of any order) and its dynamics is determined by the roots and their types to some polynomial potential (in fact, being a Hamiltonian of the reduced system).

This dynamics is completely the same as in the case of a similar bifurcation for a Hamiltonian system, which was studied and related bifurcation diagrams were plotted by L. Yu. Glebsky and L. M. Lerman [Chaos 5, 424-431 (1995; Zbl 0952.37021)] (it’s funny that the authors mention this paper in their reference list but did not point out that their results for the Hamiltonian case were obtained there). After this theoretical analysis they simulate the model equation, the generalised Swift-Hohenberg equation (which just gives a Hamiltonian reversible system), to understand how homoclinic solutions to a saddle-focus equilibrium arise in nonintegrable setting through a heteroclinic connection between the saddle-focus and a saddle periodic orbit.

This dynamics is completely the same as in the case of a similar bifurcation for a Hamiltonian system, which was studied and related bifurcation diagrams were plotted by L. Yu. Glebsky and L. M. Lerman [Chaos 5, 424-431 (1995; Zbl 0952.37021)] (it’s funny that the authors mention this paper in their reference list but did not point out that their results for the Hamiltonian case were obtained there). After this theoretical analysis they simulate the model equation, the generalised Swift-Hohenberg equation (which just gives a Hamiltonian reversible system), to understand how homoclinic solutions to a saddle-focus equilibrium arise in nonintegrable setting through a heteroclinic connection between the saddle-focus and a saddle periodic orbit.

Reviewer: Lev Lerman (Nizhny Novgorod)

##### MSC:

37G20 | Hyperbolic singular points with homoclinic trajectories in dynamical systems |

34C23 | Bifurcation theory for ordinary differential equations |

37G05 | Normal forms for dynamical systems |

37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |

70H05 | Hamilton’s equations |

##### Keywords:

homoclinic solutions; heteroclinic solutions; normal form; bifurcation; reversible Hopf bifurcation; generalized Swift-Hohenberg equation##### Software:

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\textit{P. D. Woods} and \textit{A. R. Champneys}, Physica D 129, No. 3--4, 147--170 (1999; Zbl 0952.37009)

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