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

Abstract

This paper considers an unfolding of a degenerate reversible 1–1 resonance (or Hamiltonian–Hopf) bifurcation for four-dimensional systems of time reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient ( q 2=0) that determines whether the bifurcation is super- or subcritical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur. J. Mech. B/Fluids, 15 (1996) 367–393), given a sign of the coefficient of a higher-order term ( q 4<0). Here, their analysis is generalised to include the other sign of q 4, motivated by a fourth-order ODE whose solutions model localised buckling of struts and steady states of the generalised Swift–Hohenberg equation. Numerical experiments are undertaken to determine the global behaviour of homoclinic orbits to the origin in the example which is both reversible and Hamiltonian. The normal form coefficients are calculated explicitly and a region of parameter space found where q 4>0 and −1≪ q 2<0. The normal form then shows a small-amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit. A genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies. This is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies. At each limit point the homoclinic orbit generates another ‘bump’ close to the periodic orbit. As q 2 is further decreased from zero, the snake widens until, at a critical value, a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium (a ‘kink’). For q 2 less than this value the kinks, heteroclinic tangencies and snake-like curves no longer occur.