Narrow Results

We introduce the notion of (f, δ)-recurrence for the so-called (f, δ)-processes. We show that if a function of type 2^∞ has no attractive periodic points of periods greater than 2^N for some positive integer N and has an infinite ω-limit set, then each point belonging to its infinite ω-limit set is (f, δ)-recurrent provided that δ is small enough.

We generalize some properties of (f, δ)-recurrence introduced in [Szała, 2013] to n-dimensional case and show that (f, δ)-recurrence and uniform (f, δ)-recurrence are equivalent. We also study properties of one-dimensional nonautonomous dynamical systems with randomly perturbed trajectories.

We consider the averaging principle for deterministic and stochastic perturbations of multidimensional dynamical systems for which coordinates can be introduced in such a way that the "fast" coordinates change in a torus (for Hamiltonian systems, "action-angle coordinates"). Stochastic perturbations of the white-noise type are considered. Our main assumption is that the set of action values for which the frequencies of the motion on corresponding tori are rationally dependent (and so the motion reduces to a torus of smaller dimension) has Lebesgue measure zero. Our results about stochastic perturbations imply some new results for averaging of purely deterministic perturbations.

We consider random perturbations of some one-dimensional map S : [0, 1] → [0, 1] such that parametrized by 0 < ε < 1, where {C_n} is an i.i.d. sequence. We prove that this random perturbation is small with respect to the noise level 0 < ε < 1 and give a class of one-dimensional maps for which there always exists a smooth invariant probability measure for the Markov process {X_n}_n≥0.

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for $C^1$ partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider $C^1$ diffeomorphism $f$ with dominated splitting $TM=E^{cu}\oplus E_1^c\oplus \cdots \oplus E_k^c\oplus E^{cs}$ such that $dim{E}_i^c=1$ for every $1\le i\le k$, and all the Lyapunov exponents are non-negative along $E^{cu}$ and non-positive along $E^{cs}$, we prove the asymptotically random entropy expansiveness for $f$.

This contribution probes into ergodic stationary distribution for two stochastic SVELIT (susceptible-vaccinated-early latent-late latent-infective-treated) tuberculosis (TB) models to observe the impact of white noises and color noises on TB control in random environments. We first investigate the existence and uniqueness of ergodic stationary distribution (EUESD) for the autonomous SVELIT model subject to white noises via the proper Lyapunov functions, and sufficient conditions on the extinction of disease are acquired. Next, sufficient conditions for the EUESD and the extinction of disease for the SVELIT model with Markov switching are also established. Eventually, some numerical examples validate the theoretical findings. What’s more, it has been observed that higher amplitude noises may lead to the eradication of TB, which is conducive to TB control.