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Holger Dullin

University of Sydney


Cushman R., Dullin H. R., Hanssmann H., Schmidt D. S.
The $1:\pm2$ Resonance
2007, vol. 12, no. 6, pp.  642-663
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 $\pm 1/2$ 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.
Keywords: resonant oscillators, normal form, singular reduction, bifurcation, energy-momentum mapping, monodromy
Citation: Cushman R., Dullin H. R., Hanssmann H., Schmidt D. S.,  The $1:\pm2$ Resonance, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 642-663
Dullin H. R., Vu-Ngoc S.
Symplectic Invariants Near Hyperbolic-Hyperbolic Points
2007, vol. 12, no. 6, pp.  689-716
We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system. We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C.~Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.
Keywords: completely integrable systems, hyperbolic-hyperbolic point, KAM, isoenergetic non-degeneracy, vanishing twist
Citation: Dullin H. R., Vu-Ngoc S.,  Symplectic Invariants Near Hyperbolic-Hyperbolic Points, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 689-716
Davison C. M., Dullin H. R.
Geodesic Flow on Three-Dimensional Ellipsoids with Equal Semi-Axes
2007, vol. 12, no. 2, pp.  172-197
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with $SO(2)$ symmetry, and ellipsoids with three semi-axes coinciding with $SO(3)$ symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with $SO(2)$ symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with $SO(3)$ symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$.
Keywords: eodesic flow, integrable systems, symmetry, reduction, action variables
Citation: Davison C. M., Dullin H. R.,  Geodesic Flow on Three-Dimensional Ellipsoids with Equal Semi-Axes, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 172-197
Dullin H. R.
Poisson integrator for symmetric rigid bodies
2004, vol. 9, no. 3, pp.  255-564
We derive an explicit second order reversible Poisson integrator for symmetric rigid bodies in space (i.e. without a fixed point). The integrator is obtained by applying a splitting method to the Hamiltonian after reduction by the $S^1$ body symmetry. In the particular case of a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) this integrator preserves the two momentum integrals. The method is used to calculate the complicated boundary of stability near a linearly stable relative equilibrium of the Levitron with indefinite Hamiltonian.
Citation: Dullin H. R.,  Poisson integrator for symmetric rigid bodies, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 255-564
Dullin H. R., Matveev V. S., Topalov P. I.
On Integrals of the Third Degree in Momenta
1999, vol. 4, no. 3, pp.  35-44
Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.
Citation: Dullin H. R., Matveev V. S., Topalov P. I.,  On Integrals of the Third Degree in Momenta, Regular and Chaotic Dynamics, 1999, vol. 4, no. 3, pp. 35-44
Dullin H. R., Richter P. H., Veselov A. P.
Action variables of the Kovalevskaya top
1998, vol. 3, no. 3, pp.  18-31
An explicit formula for the action variables of the Kovalevskaya top as Abelian integrals of the third kind on the Kovalevskaya curve is found. The linear system of differential equations of Picard–Fuchs type, describing the dependence of these variables on the integrals of the Kovalevskaya system, is presented in explicit form. The results are based on the formula for the actions derived by S.P.Novikov and A.P.Veselov within the theory of algebro-geometric Poisson brackets on the universal bundle of hyperelliptic Jacobians.
Citation: Dullin H. R., Richter P. H., Veselov A. P.,  Action variables of the Kovalevskaya top, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 18-31
Bolsinov A. V., Dullin H. R.
On Euler Case in Rigid Body Dynamics and Jacobi Problem
1997, vol. 2, no. 1, pp.  13-25
Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth.
Citation: Bolsinov A. V., Dullin H. R.,  On Euler Case in Rigid Body Dynamics and Jacobi Problem, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 13-25

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