Heinz Hanßmann

On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials

Abstract Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. To this end we use a special set of action–angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We analyse the 2-parameter family of (arbitrary) axially symmetric cubic potentials. This family has rich dynamics, displaying all local bifurcations of co-dimension one. With the exception of six ratios of the parameter values, the dynamical behaviour close to the origin turns out to be completely determined by the normal form of order 1. We also lay the ground for a further study at the exceptional ratios.

Symmetry and Reduction for Coordinated Rigid Bodies

Motivated by interest in the collective behavior of autonomous agents, we lay foundations for a study of networks of rigid bodies and, specifically, the problem of aligning orientation and controlling relative position across the group. Our main result is the reduction of the (networked) system for the case of two individuals coupled by control inputs that depend only on relative configuration. We use reduction theory based on semi-direct products, yielding Poisson spaces that enable efficient formulation of control laws. We apply these reduction results to satellite and underwater vehicle dynamics, proving stability of coordinated behaviors such as two underwater vehicles moving at constant speed with their orientations stably aligned.

Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator

The normal form of an axially symmetric perturbation of the isotropic harmonic oscillator is invariant under a 2-torus action and thus integrable in three degrees of freedom. The reduction of this symmetry is performed in detail, showing how the singularities of the reduced phase space determine the distribution of periodic orbits and invariant 2-tori in the original perturbation. To illustrate these results a particular quartic perturbation is analysed. AMS classification scheme numbers: 34C20, 58F05, 58F30, 70H33, 70J05, 85A05

On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter λ adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.

The reversible umbilic bifurcation

Abstract Hamiltonian systems with several degrees of freedom regularly lead to the investigation of bifurcating equilibria in reduced one-degree-of-freedom systems. This paper concerns equilibria with vanishing linearisation, a co-dimension two phenomenon in the reversible context. Under appropriate transversality conditions such equilibria have versal unfoldings related to the elliptic and hyperbolic umbilic catastrophes. This has applications to gyrostat motion and also helps to explain the dynamics defined by the normal form of the Henon-Heiles system. The occurring unfoldings turn out to be versal even in the general reversible context of not necessarily Hamiltonian systems.

The 1:±2 resonance

On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In particular we show that for the indefinite case (1:−2) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the 1:−2 resonance.

A Degenerate Bifurcation In The Hénon-Heiles Family

The normalised Hénon–Heiles family exhibits a degenerate bifurcation when passing through the separable case ‘β = 0’. We clarify the relation between this degeneracy and the integrability at β = 0. Furthermore we show that the degenerate bifurcation carries over to the Hénon–Heiles family itself.

On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems

Starting from Poincare’s fundamental problem of dynamics, we consider perturbations of integrable Hamiltonian systems in the neighbourhood of resonant Lagrangean (i.e. maximal) invariant tori with a single (internal) resonance. Applying KAM Theory and Singularity Theory we investigate how such a torus disintegrates when the action variables vary in the resonant surface. For open subsets of this surface the resulting lower dimensional tori are either hyperbolic or elliptic. For a better understanding of the dynamics, both qualitatively and quantitatively, we also investigate the singular tori and the way in which they are being unfolded by the action variables. In fact, if N is the number of degrees of freedom, singularities up to co-dimension N − 1 cannot be avoided. In the case of Kolmogorov non-degeneracy the singular tori are parabolic, while under the weaker non-degeneracy condition of Russmann the lower dimensional tori may also undergo e.g. umbilical bifurcations. We emphasize that this application of Singularity Theory only uses internal (or distinguished) parameters and no external ones.

Quasi-periodic bifurcations in reversible systems

Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parameterized by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.

A Survey on Bifurcations of Invariant Tori

Invariant tori of dynamical systems occur both in the dissipative and in the conservative context. I focus here on the latter, where the tori are intrinsically parametrised by the actions y 1,…, y n conjugate to the angles x 1,…,x n on the torus. The distribution of maximal tori in a nearly integrable Hamiltonian system is governed by the invariant tori of co-dimension one. The different Cantor families of maximal tori shrink down to normally elliptic tori and are separated by the web formed by stable and unstable manifolds of normally hyperbolic tori. The lower dimensional invariant tori form Cantor families themselves, and occurring bifurcations in turn organize the distribution of normally elliptic and hyperbolic tori.