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Effective methods for recurrence solutions in delay differential equations

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Universitat de Barcelona. Departament de Matemàtiques i Informàtica

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[eng] 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.

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Tesis Doctorals - Departament - Matemàtiques i Informàtica

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Effective methods for recurrence solutions in delay differential equations