Singular perturbation problems for a system of differential-difference equations. I (Q1336328)

The a system of singularly perturbed delay differential-difference equations of the form \[ \varepsilon\dot x(t)= -Ax(t)+ g(y(t), \lambda),\;y(t)= -f(x(t- 1), y(t- 1), \lambda)\tag{1} \] is considered. The problem is to study the relation between (1) for \(\varepsilon\) small enough and the difference equation obtained formally setting \(\varepsilon= 0\), \[ y(t)= f(A^{-1} g(y(t- 1), \lambda), y(t- 1), \lambda).\tag{2} \] In the first part of the article is studied the relation between the stability and the eigenvalues of the infinitesimal generator of the linear system \[ \varepsilon\dot x(t)= -Ax(t)+ By(t),\;y(t)= Gx(t- 1)+ Fy(t- 1)\tag{3} \] and the results are applied to investigate the stability of the equilibria of the so-called ``ring cavity equation''. In the second part, a discussion of the existence of a heteroclinic orbit corresponding to the transition layers of the singularly perturbed equations (1) is done.