A normally elliptic Hamiltonian bifurcation

Abstract

The unfolding theory, to be developed below, thenThe conjugate pair of imaginary eigenvalues gives rise to a formal rotational symmetry in all unfoldings, i.e. a rotational symmetry in their Taylor series. Here the series is considered in dependence on both the phase space variables and the parameters. This is an application of Normal Form Theory, where the terms of the formal power series are changed by canonical coordinate transformations in an inductive process. The symmetry, thus obtained, enables a formal reduction to 1 degree of freedom, around an equilibrium point with a double eigenvalue 0. For similar approaches see some of the above references, also compare e.g. Arnold [6], Takens [43,44], Broer [10,11], Golubitsky and Stewart [23] and Broer and Vegter [16]. This Normal Form Theory will be presented in Section 3.In Section 2, we begin studying the 1 degree of freedom 'backbone'-problem in its own right. Since in the plane the integral curves of the systems are the level sets of the Hamiltonian functions, here the problem reduces to Singularity Theory, e.g. In Section 4 the connection is made between the planar reduction of the symmetric system and the 'backbone'-system of Section 2. It turns out that the formal integral, obtained by normalization, is a distinguished parameter in the sense of Golubitsky and Schaeffer [51] and Schecter [41], also compare Wassermann [52]. Now Singularity Theory yields new normal forms, that are polynomial of degree 3, at least in the phase-space variables. We note that here the normalizing transformations no longer need to be canonical. Technical details from Singularity Theory have been collected in an appendix (Section 7). After this we suspend, or dereduce, to the original 4-dimensional setting, so obtaining an integrable, i.e. rotationally symmetric, approximation of the original unfolding.Thus we obtain a perturbation problem, similar to e.g. Broer [10,12] or Braaksma and Broer [8]. The perturbation term is of arbitrarily high order, both in the phase space variables and the parameters. This problem is briefly addressed in Section 5.Remarks. (i) Although in the original family only generic restrictions are imposed on the lower order terms, by Normal Form Theory and Singularity Theory, the corresponding unfolding is reduced to an arbitrarily flat perturbation of a normal form, completely determined by a 1 degree of freedom system, which is polynomial of degree 3. This polynomial character Vol. 44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19]. In [46,19] the dynamics is reduced by the energy-momentum map as well as the Moser-Weinstein method. This gives information related to specific periodic solutions, that is also important for the dynamics as a whole. Instead, we just factor out a formal rotational symmetry, and so seem to get a more ...

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

“…The fact that the Hamiltonian H − (x, 0) is unfolded completely then immediately shows that there exists a curve (µ(λ), κ(λ)) in parameter space such that g(x, λ) and f (x, µ(λ), κ(λ)) have the same number of fixed points of the same type. The results about connecting orbits carry over as well: First we see from [8,2] that within the invariant plane P our procedure yields a versal unfolding of the singular system. In particular, this implies that the results about γ hom are generic.…”

“…Then the special structure of (2) and assumptions (V), (FP) imply the following properties of the vector field.…”

“…This procedure was developed by Broer et.al. [2] who obtained C ∞ -versal unfoldings of singular fixed points of one degree of freedom Hamiltonian systems in this way (see also [9] for the planar analogue of our problem). In our (4-dimensional) case we cannot prove that the procedure yields versal unfoldings.…”

“…The main idea for the procedure yielding both the normal form of the singular system and the respective unfolding is to * Department of Mathematics, TU Ilmenau, PSF 100565, 98684 Ilmenau, Germany, Email: murmel@mathematik.tu-ilmenau.de, Phone: 0049 3677 693254, Fax: 0049 3677 693270 exploit the Hamiltonian structure and to apply Catastrophe Theory to obtain an unfolded normal form of the Hamiltonian. This procedure was successfully applied to planar Hamiltonian systems in [2,9]. The outcome in our case are two unfoldings that differ in signs of higher order terms.…”

Superharmonic instability, homoclinic torus bifurcation and water-wave breaking

“…First, any symmetry-based solution which has a maximum of the energy-momentum has a pair of eigenvalues (of the linear stability problem) passing through zero -this is Saffman's Theorem for SH instability. Secondly, any Hamiltonian system with a pair of eigenvalues passing through the origin always has a homoclinic bifurcation (see § 4 of Chapter 7 in Arnold et al 1993 andBroer et al 1993) as long as the coefficient of the first nonlinear term (i.e. the coefficient b(c) in (3.3)) is non-zero.…”