Open AccessOn polynomial forms of nonlinear functional differential equations
Author(s) -
Olivier Hénot
Publication year - 2021
Publication title -
journal of computational dynamics
Language(s) - English
Resource type - Journals
eISSN - 2158-2505
pISSN - 2158-2491
DOI - 10.3934/jcd.2021013
Subject(s) - mathematics , nonlinear system , differential equation , polynomial , embedding , eigenvalues and eigenvectors , functional equation , mathematical analysis , invariant (physics) , pure mathematics , physics , computer science , quantum mechanics , artificial intelligence , mathematical physics
In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.