We deene point attractors and set attractors for random dynamical systems via convergence in probability forward in time. This deenition of a set attractor is weaker than the usual one via almost sure pullback convergence. We derive basic properties of our weak random attractors such as uniqueness, support of invariant measures and invari-ance under (random) coordinate transformations. The noti...

Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac’s theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with...

The existence of a pullback attractor is established for the singularly perturbed FitzHughNagumo system defined on the entire space Rn when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback...

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of R . Under the assumption that the obstacle K is a closed convex and bounded subset of R with smooth boundary or it is a closed n-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In part...

To study the effects of immigration of genes (possibly transgenic) into a natural population, a one-island selection-migration model with density-dependent regulation is used to track allele frequency and population size. The existence and uniqueness of a polymorphic genetic equilibrium is proved under a general assumption about dominance in fitnesses. Also, conditions are found which guarantee...