Narrow Results

This paper deals with the qualitative analysis of a class of bilinear systems of equations describing the dynamics of individuals undergoing kinetic (stochastic) interactions. A corresponding evolution problem is formulated in terms of integro-differential (nonlocal) system of equations. A general existence theory is provided. Under the assumption of periodic boundary conditions and the interaction rates expressed in terms of convolution operators two classes of equilibrium solutions are distinguished. The first class contains only constant functions and the second one contains some nonconstant functions. In the scalar case (one equation) under suitable scaling, related to the shrinking of interaction range of each individual, the limit to the corresponding "macroscopic" equation is studied. The limiting equation turns out to be the (nonlinear) porous medium equation.

Well-balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties.

This paper is devoted to the biological equilibrium of a stochastic nonlocal PDE population model with state-selective delay. By an improved dissipativity method and a delicate analysis of interaction of stochastic terms and delay terms, we obtain a unique equilibrium, which mixes exponentially. In order to check the validity of the model, we also investigate its stochastic stability.

This paper introduces two types of Lorenz-like three-dimensional quadratic autonomous chaotic systems with 7 and 8 new parameters free of choice, respectively. Both systems are investigated at the equilibriums to study their chaotic characteristics. We focus our attention on the second type of the introduced system which consists of three nonlinear quadratic equations. Predictably, coordinates of the equilibriums are prohibitively complex. Therefore, instead of directly analyzing their stability, we prove the asymptotical characterization of equilibriums by utilizing our preliminary results derived for the first type of system. Our result shows that, though the coordinates of equilibriums satisfy a ternary quadratic, the system still contains only three equilibriums in circumstances of chaos. Sufficient conditions for the chaotic appearance of systems are derived. Our results are further verified by numerical simulations and the maximum Lyapunov exponent for several examples. Our research takes a first step in investigating chaos in Lorenz-like dynamic systems with strengthened nonlinearity and general forms of parameters.

This paper studies the problem of accumulating heterogeneous capital goods in an economy with imperfect markets populated by boundedly rational agents. It relaxes classical assumptions about information and cognition. The agents are not capable of computing an equilibrium path to steady state. Agents discover prices by interacting with each other. The economy accumulates a near-optimal mix of capital goods. The structure of interactions between agents filters their behavior in such a way that limited rationality at the micro-level does not translate to grossly inefficient outcomes at the macro-level.

In this paper, we investigate the dynamic behavior of an HIV model with stochastic perturbation. Firstly, in ODE model, the disease-free equilibrium E⁰ is globally asymptotically stable if the basic reproductive number R₀ < 1. When R₀ > 1, the endemic equilibrium E* is globally asymptotically stable. Secondly, the criterion for robustness of the system is established under stochastic perturbations. The conditions of stochastic stability of the endemic equilibrium E* are obtained. Finally, we simulate our analytical results.

In this paper, we investigate the models of the impulsive cellular neural network with generalized constant piecewise delay (IDEGPCD). To guarantee the existence, uniqueness and global exponential stability of the equilibrium state, several new adequate conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEGPCD’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of the proposed results. We end the paper with a brief conclusion.