Publications (2)0 Total impact
ABSTRACT: We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.
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 n th 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. PACS: 03.20.+i; 05.45.+b Keywords: Symplectic maps; Variational principle; Minimal periodic orbits; Stability; 1 Introduction We consider a discrete Lagrangian system on the configuration space Q, of dimension d. A discrete Lagrangian, L(x; x 0 ), x; x 0 2 Q, is a generating function for a symplectic map (x 0 ; y 0 ) = F (x; y) on Q Theta R d , that...
University of Colorado at Boulder