Symmetric Ordinary Differential Equations
by
Michal Feckan
Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynska dolina, 842 48 Bratislava, Slovakia

Let S : Rⁿ→ Rⁿ be a linear mapping such that S^p=I for a p ∈ N. We study both forced o.d.eqns [x\dot]=f(x, t) and autonomous ones [x\dot]=g(x) with properties f(Sx, t)=Sf(x, t), f(x, t+T)=f(x, t) and g(Sx)=Sg(x) for any x, t and some T > 0, respectively. We survey several results on the existence of T-S-symmetric solutions of the above o.d.eqns, i.e. solutions satisfying x(t+T)=Sx(t) ∀t ∈ R. Note such x(t) are pT-periodic in t. When g(x) is in addition globally Lipschitz continuous, we find lower bounds for all T > 0 of any nonconstant T-S-symmetric solution of [x\dot]=g(x).

Date received: June 7, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caum-78.