WIAS Preprint No. 270, (1996)

Delayed exchange of stabilities in singularly perturbed systems



Authors

  • Nefedov, Nikolai N.
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 34D15 34El5

Keywords

  • Singular perturbation, delayed loss of stability, delayed exchange of stabilities, upper and lower solution

Abstract

We consider a scalar nonautonomous singularly perturbed differential equation whose degenerate equation has two solutions which intersect at some point. These solutions represent families of equilibria of the associated equation where at least one of these families loses its stability at the intersection point. We study the behavior of the solution of an initial value problem of the singularly perturbed equation in dependence on the small parameter. We assume that the solution stays at the beginning near a stable branch of equilibria of the associated system where this branch loses its stability at some critical time tc. By means of the method of upper and lower solutions we determine the asymptotic delay t* of the solution for leaving the unstable branch. The obtained result holds for the case of transcritical bifurcation as well as for the case of pitchfork bifurcation. We consider some examples where we prove that a well-known result due to N.R. Lebovitz and R.J. Schaar about an immediate exchange of stabilities cannot be applied to singularly perturbed systems whose right hand side depends on ε.

Appeared in

  • Z. Angew. Math. Mech., 78 (1998), Suppl. 1, pp. S199-S202

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