Bifurcations into pathology for Hamiltonian systems

Hopf bifurcations for Near-Hamiltonian Systems

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian ...

Transcritical bifurcations in nonintegrable Hamiltonian systems.

We report on transcritical bifurcations of periodic orbits in nonintegrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians ...

Extended Abstract: Bifurcations in Hamiltonian Systems with Symmetry

This is an extended a.bstract for the lecture of J .E. Marsden to be presented at the ENOC-93 conference, Hamburg, August, 1993. The goal of the lecture is to outline some recent work of the lecturer, M. Golubitsky,1. Stewart, E. Knobloch, A. Mahalov and T. Ratiu on bifurcations in Hamiltonian systems with symmetry. We will survey some earlier work with D. Lewis, T. Ratiu and others to set the ...

Bifurcations of Normally Parabolic Tori in Hamiltonian Systems

Singularities of Poisson structures and Hamiltonian bifurcations

Consider a Poisson structure on C∞(R3,R) with bracket {, } and suppose that C is a Casimir function. Then {f, g} =< ∇C, (∇g × ∇f) > is a possible Poisson structure. This confirms earlier observations concerning the Poisson structure for Hamiltonian systems that are reduced to a one degree of freedom system and generalizes the Lie-Poisson structure on the dual of a Lie algebra and the KKS-symple...