We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of “stable” and “unstable” directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.
This paper studies a two-strain SIS epidemic model with a competing mechanism and a saturating incidence rate
on complex networks. This type of incidence rate could be used to reflect the crowding effect of the
infective individuals. We first obtain the associated reproduction numbers for each of the two strains which
determine the existence of the boundary equilibria. Stability of the disease-free and boundary equilibria are
further examined. Besides, we also show that the two competing strains can coexist under certain conditions.
Interestingly, the saturating incidence rate can have specific effects on not only the stability of the boundary equilibria,
but also the existence of the coexistence equilibrium. Numerical simulations are presented to support the theoretical results.
This paper studies the dynamics of a network-based SIS epidemic model with nonmonotone incidence rate.
This type of nonlinear incidence can be used to describe the psychological effect of certain diseases spread
in a contact network at high infective levels. We first find a threshold value for the transmission rate. This
value completely determines the dynamics of the model and interestingly, the threshold is not dependent
on the functional form of the nonlinear incidence rate. Furthermore, if the transmission rate is less than or
equal to the threshold value, the disease will die out. Otherwise, it will be permanent. Numerical experiments
are given to illustrate the theoretical results. We also consider the effect of the nonlinear incidence on the
Luis BarreiraDepartamento de Matemática, Instituto Superior Técnico, Universidade de LisboaJinjun LiSchool of Mathematics and Statistics, Minnan Normal UniversityClaudia VallsDepartamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa
Dynamical SystemsAlgebraic Topology and General Topologymathscidoc:1701.11013
For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.