Periodic point

Abstract: Publisher Summary Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold The totality of the solutions of the Kadomtsev– Petviashvili equation as well as of its multicomponent generalization forms an infinite dimensional Grassmann manifold. In this picture, the time evolution of a solution is interpreted as the dynamical motion of a point on this manifold. A generic solution corresponds to a generic point whose orbit (in the infinitely many time variables) is dense in the manifold, whereas degenerate solutions corresponding to points bound on those closed submanifolds that are stable under the time evolution describe the solutions to various specialized equations, such as KdV, Boussinesq, nonlinear Schrodinger, and sine-Gordon.

Abstract: We examine bifurcation phenomena for continuous one-dimensional maps that are piecewise smooth and depend on a parameter μ. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is smooth. As the parameter μ is varied, a fixed point (or periodic point) Eμ may cross the point c, and we may assume that this crossing occurs at μ=0. The investigation of what bifurcations occur at μ=0 reduces to a study of a map fμ depending linearly on μ and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Eμ may cross the point c, and for μ 0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as μ→0. More generally, for every integer m≥2, bifurcations from a fixed point attractor to a period-m attractor, a 2m-piece chaotic attractor, an m-piece chaotic attractor, or a one-piece chaotic attractor can occur for piecewise smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a, b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.

Abstract: We show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric onS2 with positive Gaussian curvature has infinitely many distinct closed geodesics.