Shadowing lemma for flows

The Shadowing Lemma for Elliptic PDE

A shadowing lemma for abelian Higgs vortices

We use a shadowing-type lemma in order to analyze the singular, semilinear elliptic equation describing static self-dual abelian Higgs vortices. Such an approach allows us to construct new solutions having an infinite number of arbitrarily prescribed vortex points. Furthermore, we obtain the precise asymptotic profile of the solutions in the form of an approximate superposition rule, up to an e...

Pseudoholomorphic Curves and the Shadowing Lemma

1. Introduction. Let M be a compact smooth manifold of dimension n, and let T * M → τ M be its cotangent bundle. T * M carries a canonical 1-form θ, which in canonical coordinates (q i , p i) is given by θ = p i dq i. Then ω := dθ is a symplectic form on T * M. To a smooth Hamiltonian H ∈ C ∞ (S 1 ×T * M, R), 1-periodic in time, we associate the Hamiltonian system ˙ x = X H (t, x), (HS) where t...

Continuous and Inverse Shadowing for Flows

We define continuous and inverse shadowing for flows and prove some properties. In particular, we will prove that an expansive flow without fixed points on a compact metric space which is a shadowing is also a continuous shadowing and hence an inverse shadowing (on a compact manifold without boundary).

Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinitedimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional ...