Premium
Discretizing hyperbolic periodic orbits of delay differential equations
Author(s) -
Farkas G.
Publication year - 2003
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200310003
Subject(s) - discretization , mathematics , periodic orbits , mathematical analysis , mathematical proof , invariant (physics) , delay differential equation , operator (biology) , orbit (dynamics) , computer assisted proof , differential equation , mathematical physics , geometry , biochemistry , chemistry , repressor , aerospace engineering , transcription factor , engineering , gene
In this paper we generalize the results on discretizing exponentially stable periodic orbits of delay differential equations obtained in [9] to the case of unstable hyperbolic periodic orbits. Moreover, we prove persistence of local invariant manifolds around the periodic orbit and a shadowing theorem which allows us to compare the dynamics of the original solution operator and its discretization in a neighborhood of the periodic orbit. The proofs use ideas from [1] but our assumptions on the closeness of the solution operator and its discretization — motivated by concrete numerical applications — are weaker. To overcome the difficulty caused by this weaker closeness assumption we use the smoothing effect of delay equations.