The cubic polynomial differential systems with two circles as algebraic limit cycles

In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.The first author is partially supported by a MINECO grant number MTM2014-53703-P, and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant 2014SGR 568, and two grants FP7-PEOPLE-2012-IRSES numbers 316338 and 318999. The third author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013

Stability of delay equations via Lyapunov functions

AbstractThe importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v′=L(t)vt, we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations

Robustness of nonuniform exponential trichotomies in Banach spaces

AbstractWe establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x′=A(t)x in Banach spaces. We also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy. We consider both trichotomies in semi-infinite intervals and trichotomies in R