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On asymptotically equivalent shallow water wave equations
H. R. Dullin, G. A. Gottwald, D. D. Holm
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…
Vanishing Twist near Focus-Focus Points
Holger R. Dullin, Vu Ngoc San
We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-…
Vanishing Twist in the Hamiltonian Hopf Bifurcation
Holger R. Dullin, Alexey V. Ivanov
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 p…
(Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation
Holger R. Dullin, Alexey V. Ivanov
The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and 1:2 (period-doubling). T…