Boele Braaksma

On a quasi-periodic Hopf bifurcation

Abstract In this paper we study quasi-periodic Hopf bifurcations for the model problem of a quasi-periodically forced oscillator, where the frequencies remain fixed. For this purpose we first consider Stoker’s problem for small damping.

Transseries for a class of nonlinear difference equations

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ŷ 0 we associate with it a stable manifold of solutions with asymptotic expansion ŷ 0. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ŷ 0 in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]Given a nonlinear analytic difference equation of level 1 with a formal power series solution ŷ 0 we associate with it a stable manifold of solutions with asymptotic expansion ŷ 0. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ŷ 0 in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]

Existence of Quasipattern Solutions of the Swift-Hohenberg Equation

We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle

MULTISUMMABILITY OF FORMAL POWER-SERIES SOLUTIONS OF NONLINEAR MEROMORPHIC DIFFERENTIAL-EQUATIONS

{\pi/q, q \geqq 4}

Toward a quasi-periodic bifurcation theory

. We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.