Accepted Paper

Inserted: 15 dec 2010
Last Updated: 15 apr 2011

Journal: Discrete Contin. Dyn. Syst.
Year: 2010

Abstract:

In this paper, we study the limit, as $\varepsilon$ goes to zero, of a particular solution of the equation ${\varepsilon}^2A{\ddot u}^{\varepsilon}(t)+\varepsilon B{\dot u}^{\varepsilon}(t)+\nabla_xf(t,u^{\varepsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\varepsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

Keywords: singular perturbations, perturbation methods, discrete approximations, vanishing viscosity, saddle-node bifurcation, heteroclinic solutions

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