Central manifolds, normal forms
نویسنده
چکیده
We consider differentiable dynamical systems generated by a diffeomorphism or a vector field on a manifold. We restrict to the finite-dimensional case, although some of the ideas can also be developed in the general case [21]. We also restrict to the behavior near a stationary point or a periodic orbit of a flow. Let the origin 0 of R be a stationary point of a C vector field X, i.e. X(0) = 0. We consider the linear approximation A = dX(0) of X at 0 and its spectrum σ(A), which we decompose as σ(A) = σs ∪ σc ∪ σu where σs resp. σc resp. σu consists of those eigenvalues with real part < 0 resp. = 0 resp. > 0. If σc = ∅ then there is no central manifold, and the stationary point 0 is called hyperbolic. Let Es, Ec and Eu be the linear A-invariant subspaces
منابع مشابه
SINGULAR LAGRANGIAN MANIFOLDS and SEMI-CLASSICAL ANALYSIS
Lagrangian submanifolds of symplectic manifolds are very central objects in classical mechanics and microlocal analysis. These manifolds are frequently singular (integrable systems, bifurcations, reduction). There has been a lot of works on singular Lagrangian manifolds initiated by Arnold, Givental and others. The goal of our paper is to extend the classical and semi-classical normal forms of ...
متن کاملNormal Forms of Vector Fields on Poisson Manifolds
We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.
متن کاملOn some generalized recurrent manifolds
The object of the present paper is to introduce and study a type of non-flat semi-Riemannian manifolds, called, super generalized recurrent manifolds which generalizes both the notion of hyper generalized recurrent manifolds [A.A. Shaikh and A. Patra, On a generalized class of recurrent manifolds, Arch. Math. (Brno) 46 (2010) 71--78.] and weakly generalized recurrent manifolds ...
متن کاملLie-point Symmetries and Nonlinear Dynamical Systems (symmetry and Approximate Symmetries of Nonlinear Equations: Bifurcations, Center Manifolds, and Normal Form Reduction)
Nonlinear symmetries of nite dimensional dynamical systems are related to nonlinear normal forms and center manifolds in the neighbourhood of a singular point. Certain abstract results can be used algorith-mically to construct the normal forms and/or the center manifold up to a given order in the perturbation expansion. We also argue that for this task, approximate symmetries are as useful as e...
متن کاملStatistical cosymplectic manifolds and their submanifolds
In this paper, we introduce statistical cosymplectic manifolds and investigate some properties of their tensors. We define invariant and anti-invariant submanifolds and study invariant submanifolds with normal and tangent structure vector fields. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal...
متن کاملInvariant manifolds in dissipative dynamical systems ∗
Invariant manifolds like tori, spheres and cylinders play an important part in dynamical systems. In engineering, tori correspond with the important phenomenon of multi-frequency oscillations. Normal hyperbolicity guarantees the robustness of these manifolds but in many applications weaker forms of hyperbolicity present more realistic cases and interesting phenomena. We will review the theory a...
متن کامل