Homoclinic orbits on invariant manifolds of a functional differential equation
The authors consider general linear functional-differential equations. In section 2 generalized exponential dichotomies are discussed. For a fixed generalized exponential dichotomy, the existence of corresponding invariant manifolds is obtained in section 3. Section 4 is devoted to a special class of invariant manifolds, called weak hyperbolic manifolds. In section 5 the authors apply their general results to the following delay equation \[ x'(t)= (\text{cosh }1)(\text{sinh }1)^{-1} x(t)- (\text{sinh }1)^{-1}(1+ x_2(t)) x(t- 1),\tag{1} \] considered by Lin, which has the homoclinic solution \[ x(t)= (\text{sinh }1)(\text{cosh }t)^{-1}.\tag{2} \] They prove that the problem of the homoclinic orbit for (1) is a finite-dimensional one, i.e. the orbit of (2) lies in a finite-dimensional invariant manifold.
scientific article; zbMATH DE number 1525503