On polynomial forms of nonlinear functional differential equations

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.

Constructive proofs for localised radial solutions of semilinear elliptic systems on Rd

“…If the nonlinearity N appearing in ( 4) is non-polynomial and involves elementary functions (namely, exponentials, logarithms, algebraic functions or compositions thereof), then a change of coordinates may be introduced to transform (4) into a higher dimensional system of polynomial ODEs (e.g. see [36][37][38]). The approach proposed in the present paper may be readily adapted, although with some additional work in extending the construction of the centre-stable manifold.…”