Effective methods for recurrence solutions in delay differential equations

Abstract

This thesis deals with the jet transport for numerical integrators and the effective invariant object computation of delay differential equations.

Firstly we study how automatic differentiation (AD) affects when they are applied to numerical integrators of ordinary differential equations (ODEs). We prove that the use of AD is exactly the same as considering the initial ODE and add new equations to the calculation of the variational flow up to a certain order.

With this result we propose to detail the effective computation when these equations are affected by a delay. In particular, the computation of the stability of equilibrium points, the computation of periodic orbits as well as their stability and continuation. Similarly the computation of quasi-orbits periodic and its stability. For such computations, we avoid the explicit generation of the Jacobian matrix and we only require the matrix-vector evaluation.

Finally, we cover the existence, uniqueness and numerical computation of the slowest direction of the stable manifold of a limit cycle of a state-dependent delay equation differential. The results are formulated in a posteriori format, which leads to rigorous proofs of numerical experiments. Specifically our result is applicable when you have a delayed perturbation and it depends on the state of an ODE in the plane.