Shadowing for linear systems of differential equations

J. Ombach
1993 Publicacions matemàtiques  
For a system of linear ordinary differential equations with constant coefficients a simple proof is given that hyperbolicity is equivalent to shadowing. 1 . The notion of shadowing or the pseudo orbit tracing property (abbr.POTP) usually appears if one considers a dynamical system on a compact manifold . The famous Shadowing Lemma says, rough1y speaking, that hyperbolicity implies the POTP. There are a number of proofs of this result : all of them rather complicated and tedious . In every case
more » ... us . In every case the compactness is essential. Morimoto, however, in [4] considered this property in R' for discrete dynamical systems generated by linear homeomorphisms . He and Kakubari in [3] proved that hyperbolicity is equivalent to the POTP for such systems . In [5] we give a different proof which covers also infinite dimensional case. In this note we show the analogous statement for systems of linear ordinary differential equations with constant coefficients . A proof that hyperbolicity implies shadowing established for discrete case in [4] may be transformed to continuous case, [7], yet we give here a different proof which is simpler and works also in discrete case. A proof of the converse statement mimics the discrete version from [5] . The concept of the POTP comes from Anosov and Bowen . For dynamical systems with continuous time it was examined by Franke and Selgrade in [1] and by Thomas in papers [8] [9] and others, see Thomas' papers for more details. For such systems the common definition of the POTP is as follows . Every S-pseudo-orbit with sufficiently small S > 0 can be arbitrarily close uniformly approximated by a true orbit after some reparametrization of time on the true orbit . What we are going to show is that for a system of differential equations *Supported by Polish scientific grant RP. 1. 10.
doi:10.5565/publmat_37293_01 fatcat:cz6rtbec7jbvpkf423mhdsyp4u