Search results for "Center manifold"
showing 4 items of 4 documents
Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity
2001
Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.
Some topological invariants for three-dimensional flows
2001
We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. (c) 2001 American Institute of Physics.
INSTABILITY OF HAMILTONIAN SYSTEMS IN THE SENSE OF CHIRIKOV AND BIFURCATION IN A NON LINEAR EVOLUTION PROBLEM EMANATING FROM PHYSICS
2004
We prove the existence of a minimal geometrico-dynamical condition to create hyperbolicity in section in the vicinity of a transversal homoclinic partially hyperbolic torus in a near integrable Hamiltonian system with three degrees of freedom. We deduce in this context a generalization of the Easton's theorem of symbolic dynamics. Then we give the optimal estimation of the Arnold diffusion time along a transition chain in the initially hyperbolic Hamiltonian systems with three degrees of freedom with a surrounding chain of hyperbolic periodic orbits .In a second part, we describe geometrically a mechanism of diffusion studied by Chirikov in a near integrable Hamiltonian system with three de…
Approximation of functions over manifolds : A Moving Least-Squares approach
2021
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated…