Mohrenstr. 39, 10117, Berlin, Germany
Weierstrass Institute for Applied Analysis and Stochastics
Nefedov N. N., Recke L., Schneider K. R.
Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems
2010, vol. 15, no. 2-3, pp. 382-389
We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein–Rutman theorem.
Gonchenko S. V., Schneider K. R., Turaev D. V.
Quasiperiodic regimes in multisection semiconductor lasers
2006, vol. 11, no. 2, pp. 213-224
We consider a mode approximation model for the longitudinal dynamics of a multisection semiconductor laser which represents a slow-fast system of ordinary differential equations for the electromagnetic field and the carrier densities. Under the condition that the number of active sections $q$ coincides with the number of critical eigenvalues we introduce a normal form which admits to establish the existence of invariant tori. The case $q=2$ is investigated in more detail where we also derive conditions for the stability of the quasiperiodic regime