We discuss a stochastic predator-prey system with Holling II functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we deduce the conditions that there is a stationary distribution of the system, which implies that the system is permanent. At last, we give the conditions for the system that is going to be extinct.

Letting $(X,d)$ be a metric space, $f:X\to X$ a continuous map, and $(ℱ(X),D)$ the space of nonempty fuzzy compact subsets of $X$ with the Hausdorff metric, one may study the dynamical properties of the Zadeh's extension $\widehat{f}:ℱ(X)\to ℱ(X):u\mapsto \widehat{f}u$. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations between $f$ and $\widehat{f}$. More specifically, we study the transitivity, weakly mixing, periodic density in system $(X,f)$, and its connections with the same ones in its fuzzified system.

This paper deals with the problem of stability for a class of Lur’e systems with interval time-varying delay and sector-bounded nonlinearity. The interval time-varying delay function is not assumed to be differentiable. We analyze the global exponential stability for uncertain neutral and Lur’e dynamical systems with some sector conditions. By constructing a set of improved Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, we establish some stability criteria in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the results.

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2$g$, where $g>1$ is an integer and the discriminant of such fields has only two prime divisors.

Let $T_1$ and $T_2$ be piecewise smooth circle homeomorphisms with break points and identical irrational rotation numbers. We provide one sufficient and necessary condition for the absolute continuity of conjugation map between $T_1$ and $T_2$.

The stabilization of dynamic switched control systems is focused on and based on an operator-based formulation. It is assumed that the controlled object and the controller are described by sequences of closed operator pairs $\mathrm{(L,C)}$ on a Hilbert space $H$ of the input and output spaces and it is related to the existence of the inverse of the resulting input-output operator being admissible and bounded. The technical mechanism addressed to get the results is the appropriate use of the fact that closed operators being sufficiently close to bounded operators, in terms of the gap metric, are also bounded. That philosophy is followed for the operators describing the input-output relations in switched feedback control systems so as to guarantee the closed-loop stabilization.