Abstract. Existence of homoclinic orbits in the cubic nonlinear Schrödinger equation under singular perturbations is proved. Emphasis is placed upon the regularity of the semigroup e ǫt∂
Recent experimental and theoretical studies on the magnetization dynamics
driven by an electric current have uncovered a number of unprecedented rich
dynamic phenomena. We predict an intrinsic chaotic dynamics that has not been
previously anticipated. We explicitly show that the transition to chaotic
dynamics occurs through a series of period doubling bifurcations. In chaotic
regime, two dramatically different power spectra, one with a well-defined peak
and the other with a broadly distributed noise, are identified and explained
Abstract. First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H ( = 0, 1, 2, · · · ). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits (ν → 0 + ) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-stable manifold of a steady state are O( √ ν), while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are O(ν), as ν → 0 + . Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete estimate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other two examples, partial estimates on the eigenvalues are obtained.
This is a rather comprehensive study on the dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a). zero viscosity limit of the spectra of linear Navier-Stokes operator, (b). heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Besides Navier-Stokes and Euler equations, we also study two models of them.
The work [1] is generalized to the singularly perturbed nonlinear Schrödinger (NLS) equation of which the regularly perturbed NLS studied in [1] is a mollification. Specifically, the existence of Smale horseshoes and Bernoulli shift dynamics is established in a neighborhood of a symmetric pair of Silnikov homoclinic orbits under certain generic conditions, and the existence of the symmetric pair of Silnikov homoclinic orbits has been proved in [2]. The main difficulty in the current horseshoe construction is introduced by the singular perturbation ǫ∂ 2x which turns the unperturbed reversible system into an irreversible system. It turns out that the equivariant smooth linearization can still be achieved, and the Conley-Moser conditions can still be realized. Contents 1. Introduction 225 2. Equivariant Smooth Linearization 227 3. The Poincaré Map and Its Representation 228 4. The Fixed Points of the Poincaré Map P 230 5. Existence of Chaos 231 6. Numerical Evidence for the Generic Assumptions 236 References 237
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