Non-autonomous periodic perturbations of a nonlinear delayed system near an equilibrium

P. AMSTER

Sao Paulo

Workshop; South American Workshop on Integral and Differential Equations; 2018

Let Omega be a smooth bounded domain in R^N and consider the autonomous systemof delay differential equationsu´(t)=g(u(t),u´(t-au)) where g : R^No R^N is continuous. Let e be an equilibrium point and assume that the linearisation at e has no nontrivial T-periodic solutions, then a standard argument shows that the small non-autonomous perturbations of (1) admit at least one T-periodic solution. Furthermore, if the field G(x) := g(x, x) is inwardly pointing over Omega and the perturbation is small, then for the number of T-periodic solutions isgenerically at least |Gamma ± 1| + 1, where Gamma is the Euler characteristic Omega. In this talk we shall give a topological proof of this fact and related results.