Local Invariant Manifolds Research Articles

The poincaré manifold at a stable equilibrium of c2planar flow

We establish the existence of a unique C2 local invariant manifold at the stable equilibrium of a C2 vector field. Here, in contrast to the standard case, the eigenvalue ratio μ/⋋ is bounded above by the degree of smoothness of the vector field.

A new method for the existence of solutions of nonlinear differential equations

Many problems is analysis reduce to solving an equation Au=f, where A is an operator on a space into another space, u is the unknown, and f is a given element. The existence of a local solution near a trivial solution u0 of Au, = 0, where 0 is near f, is usually solved by the classical implicit function theorem. As a matter of fact, the Picard iteration procedure is used. Unfortunately, this procedure cannot be applied in some problems of partial differential equations. These cases occur if the loss of derivatives appears [6, 7, 31. To get around this difficulty a more sophisticated iteration procedure is needed. Moser [ 1 ] has found such a method which enables one to handle a loss of derivatives by using a rapidly converging iteration scheme. The Nash-Moser technique is nicely described in the papers [S, lo]. We prove in this paper several theorems on the existence of solutions, with the application to several types of specific problems. The main interest is that in many cases these results can be used in situations which up to now have required the Nash-Moser implicit function theorem, yet they have classical proofs. We also apply abstract results to show the existence of local invariant manifolds for singularly perturbed ordinary differential equations and we give a simple proof of a theorem of J. Moser for invariant tori [l]. The plan of our paper is as follows. In Section 2 we present proofs of several existence theorems in an abstract setting in a scale of Hilbert spaces. The hypotheses are those commonly found in problems which have up to now required the application of the Nash-Moser technique, however, the results have classical proofs based on the degree of mappings [S]. We shall give in the proof of Theorem 2.8 a new iteration procedure which retains supergeometric convergence properties. We require obligations only for the linearization of our equation. This is the main advance of this theorem. Section 3 of the paper presents applications of the theorems of Section 2. There are demonstrated proofs of the above mentioned theorems of

Linearization principle for a toroidal Hall current plasma with viscosity and resistivity

In this paper the stability analysis of an incompressible toroidal Hall Current plasma with resistivity and viscosity on the basis of the linearization of the governing equations and boundary condition is rigorously justified. A nonlinear local existence theorem for an initial-boundary value problem is first proved, and local stable and unstable invariant manifolds of a nonlinear resolving operator are then constructed. It is shown that linear stability implies nonlinear stability and global existence, and linear instability implies nonlinear instability in some sense.

ON UNSTABLE SETS OF EVOLUTION EQUATIONS IN THE NEIGHBORHOOD OF CRITICAL POINTS OF A STATIONARY CURVE

Equations of the form are considered, for example, parabolic and hyperbolic equations. It is proved that the change of the local unstable invariant manifolds of such equations is determined by the form of the stationary curve , . Bibliography: 9 titles.

Invariant Manifolds, Quasi-Geostrophy and Initialization

Abstract A critical examination of the foundations of the slow manifold concept is presented. We study the behavior of the simple Lorenz truncated primitive equation model and show that free gravity modes exist for large forcing. For small forcing, the flow is dominated by geostrophic motion and quasi-invariant manifolds are found. However, we present arguments against the existence of a strictly invariant manifold. We show that the bounded derivative conditions are generally not satisfied, even when an invariant manifold exists; smooth functions always possess fast transients. We present a new initialization method based on the analytic expansion of local invariant manifolds and compare it with Machenhauer's method on a simple example.