Narrow Results

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.

In this paper, we study a quantum Boltzmann equation with a harmonic oscillator for isotropic gases of bosons and fermions, respectively. This model comes from physics literatures (see, e.g., [M. Holland, J. Williams and J. Cooper, Bose–Einstein condensation: Kinetic evolution obtained from simulated trajectories, Phys. Rev. A 55 (1997) 3670–3677]). The distribution function, i.e. the solution, is discrete in the energy variable. We give the classification of equilibria of the equation for bosons and fermions, respectively, and prove the global existence, uniqueness and the strong convergence to equilibrium for solutions of the equation.

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.

In the general case of the free rigid body, we will give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of , we prove that there are 15 coordinate-type Cartan subalgebras which on a regular adjoint orbit give 15 Weyl group orbits of equilibria. These coordinate-type Cartan subalgebras are the analogs of the three axes of equilibria for the classical rigid body on . The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are 10 other continuous families of equilibria.

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.

The discovery of a priori bounds for semilinear parabolic equations with superlinear nonlinearity enables the use of the parabolic semiflow in proving existence results for equilibria, as an alternative to minimax methods from the calculus of variations. We present an overview of results that use this approach, discuss its shortcomings and advantages, and outline possible future research directions.