5 research outputs found
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation
We present an algorithm for the rigorous integration of Delay Differential
Equations (DDEs) of the form . As an application, we
give a computer assisted proof of the existence of two attracting periodic
orbits (before and after the first period-doubling bifurcation) in the
Mackey-Glass equation
Rigorous Enclosures of a Slow Manifold
Slow-fast dynamical systems have two time scales and an explicit parameter
representing the ratio of these time scales. Locally invariant slow manifolds
along which motion occurs on the slow time scale are a prominent feature of
slow-fast systems. This paper introduces a rigorous numerical method to compute
enclosures of the slow manifold of a slow-fast system with one fast and two
slow variables. A triangulated first order approximation to the two dimensional
invariant manifold is computed "algebraically". Two translations of the
computed manifold in the fast direction that are transverse to the vector field
are computed as the boundaries of an initial enclosure. The enclosures are
refined to bring them closer to each other by moving vertices of the enclosure
boundaries one at a time. As an application we use it to prove the existence of
tangencies of invariant manifolds in the problem of singular Hopf bifurcation
and to give bounds on the location of one such tangency