Renormalization group second order approximation for singularly perturbed nonlinear ordinary differential equations

Abstract : We consider a two time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ε of the scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ε 2). It is a classical problem, studied using the Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in two steps. In the first step we show that the RG method allows to approximate the fast time variables by their RG expansion taken at the slow time unknowns. Next we study the slow time equations, where the fast time unknowns are replaced by their RG expansion and show the second order uniform error estimate. The procedure is computationally less demanding than the classical Vasil'eva-O'Malley expansion and allows a higher order extension of Hoppensteadt's result on the Tikhonov singular perturbation theorem for infinite times.
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Anna Marciniak-Czochra, Andro Mikelic, Thomas Stiehl. Renormalization group second order approximation for singularly perturbed nonlinear ordinary differential equations. Mathematical Methods in the Applied Sciences, Wiley, 2018, 41, pp.5691--5710. ⟨hal-01795884⟩

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