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Topology of Energy Surfaces and Singularities of Curves

By Milena Radnović and Vered Rom-kedar


The differential topology of energy surfaces of integrable systems determines much of the behavior of the solutions of the corresponding nearintegrable systems. To study this topology, one constructs the ‘bifurcation set ’ in the energy-momentum diagram. In the 2 degrees of freedom case, the bifurcation set is composed of curves and isolated points in R 2. It is proved that under natural non-degeneracy conditions, changes in the differential topology of energy surfaces of smooth 2 degrees of freedom integrable Hamiltonian systems imply that the bifurcation set must have singularities at the corresponding energy levels. The notion of the critical set, as the set of singularities of the bifurcation set, is introduced, and this set is divided to essential and non-essential parts. It is proved that generically, essential critical points appear if and only if the differential topology of the isoenergy manifolds changes at the corresponding energy levels. The proof of these two main theorems incorporates a concise description of the isoenergy surfaces near the bifurcation set and near the critical set using Fomenko graphs; thus, the structure of the energy surfaces near isolated fixed points, near circles of fixed points (normally hyperbolic or normally elliptic), near parabolic circles and near global bifurcations of separatrices of normally hyperbolic circles is completely classified and listed using Fomenko graphs. This may be viewed as completing Smale program for relating the energy surfaces structure to singularities of the energy-momentum mappings for generic integrable two degrees of freedom systems.

Year: 2009
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