On polynomial forms of nonlinear functional differential equations

Olivier Hénot

doi:10.3934/jcd.2021013

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2021, Volume 8, Issue 3: 307-323. Doi: 10.3934/jcd.2021013

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On polynomial forms of nonlinear functional differential equations

Olivier Hénot^,

McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montréal, Québec, H3A 0B9, Canada

^* Corresponding author: Olivier Hénot
^* Corresponding author: Olivier Hénot

Received: April 1, 2020

Revised: April 1, 2021

Early access: June 28, 2021

Published online: June 28, 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 34Kxx, 34K17, 34K12, 34K20, 37C25; Secondary: 37C86.

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