Stable Manifold Theorem Research Articles

Singularities in vibro-impact dynamics

The non-differentiable nature of vibro-impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems. In one dimensional periodically excited system, the breakdown leads to the “shredding” of the stable manifolds, and this process is analyzed in detail. Bodily translation of filaments of stable manifolds can lead to homoclinic intersections, and this topic is discussed in the context of a new bifurcation due to “re-entrance”. Shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets.

A stable manifold theorem for the gradient flow of geometric variational problems associated with a quasi-linear parabolic equations

On demontre le theoreme de variete stable pour une classe d'equations paraboliques quasi-lineaires et on etudie le comportement asymptotique du flot gradient des problemes variationnels geometriques

A stable manifold theorem for the Yang-Mills gradient flow

The purpose of this paper is to study the asymptotic behavior of the gradient flow of the Yang-Mills functional near Yang-Mills connections

A stable manifold theorem for the curve shortening equation

AbstractWe present a family of homothetic solutions to the equation for a planar curve and prove the existence of nonlinear stable and unstable manifolds about each such solution.

On the generic nonexistence of first integrals

The property of having no C n {C^n} first integrals other than constants is shown to be generic in Diff r ( M ) {\operatorname {Diff} ^r}(M) for each r = 1 , 2 , … r = 1,2, \ldots , where n n is the dimension of M M .

Invariant Manifolds and the Onset of Reversal in the Rikitake Two-Disk Dynamo

In this paper we prove a stable manifold theorem for noncompact solution curves of an ordinary differential equation and apply the theorem to establish the existence of two-dimensional stable and unstable manifolds for a particular solution curve of the Rikitake equations. Some information on the configuration of these surfaces in phase space is obtained yielding a rough description of the oscillatory nature of typical solutions.

Parabolic orbits in the planar three body problem

It is not known whether there exist oscillatory and capture orbits in the planar 3-body problem. Sitnikov [6] proved such orbits exist for the restricted 3-body problem. Alekseev [I] extended this work and related it to the existence of homoclinic orbits. McGehee and Easton ]3] studied a model problem closely related to the planar 3-body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3-sphere. The goal of this paper is to construct an invariant 3-sphere “at infinity” in the planar 3-body problem and to prove that the stable and unstable manifolds of this sphere are Lipschitz manifolds. To complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3-sphere intersect each other. This is a difficult question which is not treated here. However, using methods of Melnikov and using one of the masses as a perturbation parameter such an investigation might be possible. The stable manifold theorem which is proven here is not standard since the invariant 3-sphere is not hyperbolic. Geometric methods of McGehee [ 4 ] are extended and applied to obtain the theorem. A special case of the system of equations that we study has the form J, =x: dY)lX, + E,(x, Y)L f, = 4 dY>l-x* + E,(x, Y)L (0.1) 4’ =

Hammer's X -Ray Problem and the Stable Manifold Theorem

Hammer's <i>X</i> -Ray Problem and the Stable Manifold Theorem

Invariant manifolds for regular points

In this article we prove, for a differentiable vector field or a diffeomorphism on a smooth manifold, that the set of points such that the semitrajectories issuing from them approach a particular semitrajectory at a given exponential rate, constitute a differentiable submanif old, provided the differential of the flow has a certain similar behavior on that trajectory. (See Theorem 1 below, for a precise statement). In particular, the stable manifold theorem for hyperbolic sets ([3], [6, XI]) follows as a corollary. Although we only consider the Crease, the same methods, which are essentially classical ([2, Ch. XIII]), could be applied to obtain higher differentiability properties. Since I have not seen in the literature this type of results for points which are neither equilibrium nor periodic points, and on account of [6, XI-8], I thought that their publication might not be entirely devoid of interest. 1* Terminology and notation are standard. If X is a differentiable vector field on a smooth manifold M, will always denote the corresponding flow, and φt the diffeomorphism x —> φ(x, t), xeM, te R. For brevity, we shall sometimes write x(t) or y(t) instead of φ(x, t) or φ(y, t) respectively.

Ergodic theory of differentiable dynamical systems

Iff is a C1 + ɛ diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.

Morse-Smale diffeomorphisms on tori

where d is any metric on M compatible with its topology. If x is a hyperbolic periodic point of f, the stable manifold theorem[7] implies that then W”(x) and W”(x) are 1-1 immersed Euclidean spaces. An f f Diff M is Morse-Smale iff: (1) Q(f) is finite, and hence equal to the set of periodic points. (2) The periodic points are hyperbolic. (3) (Transversal intersection condition) For each pair of periodic points p, 4, W’(p) and W”(q) have transversal intersection. We will need the following special case of the fin-stability theorem due to J. Palis [8].

A Stable Manifold Theorem for a System of Volterra Integro-Differential Equations

We study the behavior of solutions of the perturbed system \[({\text{N}})\qquad z'(t)Az(t) + B * z(t) + Hz(t),\quad 0 \leqq t < \infty ,\] with initial condition $z(0) = z_0$, for ($| {z_0} |$ sufficiently small; z is an n component vector, A is a real $n \times n$ matrix, B is a real $n \times n$ matrix of functions in $L^1 (0,\infty )$, $B * z$ is the convolution, and the perturbation vector H of nonlinear functionals defined on appropriate Banach spaces represents higher order terms in z. The underlying hypothesis is that the equation $({\text{E}})\qquad \det (sI - A - \hat B(s)) = 0,\quad \operatorname{Re} s \geqq 0,$ where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{B} $ is the Laplace transform of B has N roots $\{ {s_1 ,s_2 , \cdots ,s_N } \}$, $\operatorname{Re} s_j > 0$, $1 \leqq N < \infty $, where each root $s_j $ has multiplicity $m_j $, $j = 1, \cdots ,N$, and (E) has no roots $s_j $ with $\operatorname{Re} s_j = 0$. The principal result is a stable manifold theorem for (N); the classical such theorem of H. Weyl for ordinary differential equations $(B \equiv 0)$ is a special case.

On the Stable Manifold Theorem for Discrete Time Dependent Processes in Banach Spaces

On the Stable Manifold Theorem for Discrete Time Dependent Processes in Banach Spaces

A stable manifold theorem for degenerate fixed points with applications to celestial mechanics

A stable manifold theorem for degenerate fixed points with applications to celestial mechanics

On the Stable Manifold Theorem

On the Stable Manifold Theorem