Homoclinic Orbits on Invariant Manifolds of a Functional Differential Equation

Exponential Dichotomies and Homoclinic Orbits in Functional Differential Equations*

Suppose an autonomous functional differential equation has an orbit r which is homochnic to a hyperbolic equilibrium point. The purpose of this paper is to give a procedure for determining the behavior of the solutions near r of a functional differential equation which is a nonautonomous periodic perturbation of the original one. The procedure uses exponential dichotomies and the Fredholm alter...

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Homoclinic bifurcations are important phenomena that cause global re-arrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that...

Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equation dU dt = 1 ζ(U) φs(U) + φns(U), (0.1) where U ∈ R , the functions φs ∈ R N and φns ∈ R N are of class C and ζ is a real valued C function. The equation is singular in the sense that ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighbourhood of a point Ū such that φs(Ū) = φns(Ū) = ~0, ζ(Ū) = 0. We inves...

Homoclinic orbits to invariant sets of quasi-integrable exact maps

Dynamics of Differential Equations on Invariant Manifolds

The simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice. Appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics. This paper develops an approach which avoids the selection of coordinate systems on the man...