Stability of minimal periodic orbits |
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Affiliation: | 1. Department of Mathematics, Northeastern University, USA;2. Mathematical Sciences Institute, Australian National University, Australia;3. Department of Mathematics, Yale University, and School of Mathematics, IAS, USA;4. School of Mathematics and Statistics, University of Sydney, Australia |
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Abstract: | Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map. |
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