Holger R. Dullin

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.

Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves

We derive the Camassa–Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg–de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the fifth-order integrable equation in the KdV hierarchy) by using the non-linear/non-local transformations introduced in Kodama (Phys. Lett. A 107 (1985a) 245; Phys. Lett. A 112 (1985b) 193; Phys. Lett. A 123 (1987) 276). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

Homoclinic bifurcations for the Hénon map

Abstract 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.

Generalized Hénon maps: the cubic diffeomorphisms of the plane

Abstract In general a polynomial diffeomorphism of the plane can be transformed into a composition of generalized Henon maps. These maps exhibit some of the familiar properties of the quadratic Henon map, including a bounded set of bounded orbits and an anti-integrable limit. We investigate in particular the cubic, area-preserving case, which reduces to two, two-parameter families of maps. The bifurcations of low period orbits of these maps are discussed in detail.

ACTION INTEGRALS AND ENERGY SURFACES OF THE KOVALEVSKAYA TOP

The different types of energy surfaces are identified for the Kovalevskaya problem of rigid body dynamics, on the basis of a bifurcation analysis of Poincare surfaces of section. The organization of their foliation by invariant tori is qualitatively described in terms of Poincare-Fomenko stacks. The individual tori are then analyzed for sets of independent closed paths, using a new algorithm based on Arnold’s proof of the Liouville theorem. Once these paths are found, the action integrals can be calculated. Energy surfaces are constructed in the space of action variables, for six characteristic values of energy. The data are presented in terms of color graphs that give an intuitive access to this highly complex integrable system.

Generalizations of the Stormer problem for dust grain orbits

Abstract We investigate the generalized Stormer problem, which includes electromagnetic and gravitational forces on a charged dust grain near an axisymmetric planet. For typical charge-to-mass ratios neither force can be neglected. The effects of the different forces are discussed in detail. Thus, including the gravitational force gives rise to stable circular orbits lying in a plane entirely above/below the equatorial plane. A modified third Kepler’s law for these orbits is found and analyzed.

Symbolic dynamics and periodic orbits for the cardioid billiard

The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topologically well ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov - Sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. Furthermore, the statistical properties of periodic orbits are investigated.

Stability of levitrons

Abstract The Levitron is a magnetic spinning top which can levitate in the constant field of a repelling base magnet. An explanation for the stability of the Levitron using an adiabatic approximation has been given by Berry. In experiments the top eventually loses stability at a critical spin rate which cannot be predicted by Berry’s approach. The present work develops an exact theory of the Levitron with six degrees of freedom which allows for the calculation of critical spin rates. The main result is a complete classification of possible Levitrons that allow for an interval of stable spin rates. Stability of the relative equilibrium is lost in Hamiltonian Hopf bifurcations if either the spin rate is too large or too small.

Complete Poincare sections and tangent sets

Trying to extend a local definition of a surface of a section, and the corresponding Poincare map to a global one, one can encounter severe difficulties. We show that global transverse sections often do not exist for Hamiltonian systems with two degrees of freedom. As a consequence we present a method to generate the so-called W-section, which by construction will be intersected by (almost) all orbits. Depending on the type of tangent set in the surface of the section, we distinguish five types of W-sections. The method is illustrated by a number of examples, most notably the quartic potential and the double pendulum. W-sections can also be applied to higher dimensional Hamiltonian systems and to dissipative systems.

Quantum monodromy in the two-centre problem

Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H+2 and the asymmetric case of H He++ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.