THE USE OF TIME AS A COMPLEX VARIABLE
Carles Simo
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona
carles@maia.ub.es
Abstract
Dynamical systems try to describe the behaviour of mathematical models
of the natural sciences. These models are, in the deterministic case,
ordinary differential equations, diffeomorphisms, partial differential
equations, differential delay equations, etc. Being all of them
evolution equations, to model the behaviour of the natural phenomena,
time appears as the independent variable. In this paper we review the
importance of considering time as a complex variable instead of a real
one, as it would be suggested by the physical meaning of time. The
topics discussed include bounds on the averaging theory for slow
systems, splitting of separatrices, quasiperiodic equations and
solutions, integrability of Hamiltonian systems, delay in the loss of
stability, complex homoclinic points, etc. In all cases we assume that
the models are analytic.