Vanishing Twist in the Hamiltonian Hopf Bifurcation

The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus-focus point. We explicitly calculate the frequency map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy-momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus-focus point in the image of the energy-momentum map. This is equivalent to the vanishing of twist in a Poincar\'e map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the non-linear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian Hopf bifurcation.Comment: 18 pages, 4 figure

Normal Forms for Symplectic Maps with Twist Singularities

We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-$T$ mapping of a two-degree-of freedom Hamiltonian flow. Consequently there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied, one-degree-of freedom case but is essentially nonintegrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduced this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.Comment: LaTex, 27 pages, 21 figure

On asymptotically equivalent shallow water wave equations

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

Scattering invariants in Euler’s two-center problem

The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. The analogue for volume-preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume-preserving map whose inverse is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur

Homoclinic Bifurcations for the Henon Map

Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. We use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations. We find a bound on the parameter range for which the Henon map exhibits a complete binary horseshoe as well as a subshift of finite type. We classify homoclinic bifurcations, and study those for the area preserving case in detail. Simple forcing relations between homoclinic orbits are established. We show that a symmetry of the map gives rise to constraints on certain sequences of homoclinic bifurcations. Our numerical studies also identify the bifurcations that bound intervals on which the topological entropy is apparently constant.Comment: To appear in PhysicaD: 43 Pages, 14 figure

Geometry of integrable dynamical systems on 2-dimensional surfaces

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions, and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.Comment: 31 pages, 12 figures, submitted to a special issue of Acta Mathematica Vietnamic

Phase-Space Volume of Regions of Trapped Motion: Multiple Ring Components and Arcs

The phase--space volume of regions of regular or trapped motion, for bounded or scattering systems with two degrees of freedom respectively, displays universal properties. In particular, sudden reductions in the phase-space volume or gaps are observed at specific values of the parameter which tunes the dynamics; these locations are approximated by the stability resonances. The latter are defined by a resonant condition on the stability exponents of a central linearly stable periodic orbit. We show that, for more than two degrees of freedom, these resonances can be excited opening up gaps, which effectively separate and reduce the regions of trapped motion in phase space. Using the scattering approach to narrow rings and a billiard system as example, we demonstrate that this mechanism yields rings with two or more components. Arcs are also obtained, specifically when an additional (mean-motion) resonance condition is met. We obtain a complete representation of the phase-space volume occupied by the regions of trapped motion.Comment: 19 pages, 17 figure