Rigorous enclosures of a slow manifold
2012 (English)In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 11, no 3, 831-863 p.Article in journal (Refereed) Published
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 the method 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.
Place, publisher, year, edition, pages
2012. Vol. 11, no 3, 831-863 p.
34C45, 34E13, 34E15, 37G25, 37M99
Mathematical Analysis Computational Mathematics
Research subject Mathematics with specialization in Applied Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-179185DOI: 10.1137/120861813OAI: oai:DiVA.org:uu-179185DiVA: diva2:543616
FunderSwedish Research Council, 2010-598