When fluid in a rectangular tank sits upon a platform which is oscillating with sufficient amplitude, surface waves appear in the "Faraday resonance." Scientists and engineers have done bifurcation analyses which assume that there is a center manifold theory using a finite number of excited spatial modes. We establish such a center manifold theorem for Xiao-Biao Lin’s model in which potential f...

In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...

In this paper we analyze the role of general (possibly nonlinear) time-independent Lie point symmetries in the study of nite-dimensional autonomous dynamical systems, and their relationship with the presence of manifolds invariant under the dynamical ow. We rst show that stable and unstable manifolds are left invariant by all Lie point symmetries admitted by the dynamical system. An identical r...

We consider the coupled Maxwell{Lorentz equations for a single charge with a rigid charge distribution subject to slowly varying external potentials. In the adiabatic limit, to lowest order, the motion of the charge is governed by an eeective Hamiltonian. In the next order we obtain, as a small correction, the radiation reaction containing q {terms. This equation has a stable center manifold an...

We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference H = Dm + V has only two double eigenvalues and that degeneracies are due to a symmetry of H (theorem of Kramers). In this case, we can build a small 4-dimensional manifold of stationary solutions tangent to the first eigenspace of H . Then we assume that a resona...

In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of the one dimensional center manifold of a parabolic-hyperbolic fixed point of a real-analytic map. We use the parameterization method, reducing the dynamics on the center manifold to a polynomial. We prove that the asymptotic expansions of the center manifold are of Gevrey typ...