In this paper, a new method is adopted to construct a Lyapunov function for the endemic equilibrium of the J. Mena-Lorca and H.W. Hothcote’s SIRS epidemic model with bilinear incidence and constant recruitment. On the basis of the Lyapunov function, the domain of the attraction of the endemic...

This paper investigates the asymptotic stability of a class of impulsive high-order neural networks, which can be considered as an expansion of Hopfield neural networks. By employing Lyapunov functions and linear matrix inequality (LMI) technique, sufficient conditions that guarantee the global...

The paper discusses algorithms for investigation of the stability of mechanical systems: (i) investigation of stability in the first approximation; (ii) Algorithms for constructing Lyapunov functions. A special attention is given to the cases critical in the Lyapunov sense; (iii) algorithms...

After investigating the effect of the frequency of an external electrical stimulation on the chaotic dynamics of a single FitzHugh–Nagumo (FHN) neuron, this paper derives both a sufficient and a necessary condition of the coupling coefficient for self-synchronization of two interacting FHN...

In the present paper, a class of stochastic Runge–Kutta methods containing the second order stochastic Runge–Kutta scheme due to E. Platen for the weak approximation of Itô stochastic differential equation systems with a multi-dimensional Wiener process is considered. Order 1 and order 2...

The phenomenon of ‘synchronization’ of physical diffusion is widely discussed in the physical literature. In this paper, we give a simple rigorous proof of the synchronization for a one-dimensional diffusion including the one-dimensional counterpart of a physical diffusion described by a...

We consider scalar stochastic differential equations of the formdXt=μ(Xt)dt+σ(Xt)dBt,X0=x0,where B is a standard Brownian motion. Suppose that the coefficients are such that the solution X possesses the (a, b)-invariance property for some interval (a,b)⊂R:Xt∈(a,b) for all t≥0 if...

A ‘standard’ second order weak Runge–Kutta method for a stochastic differential equation can be applied only in the case where the equation is understood in the Stratonovich sense. To adapt Runge–Kutta type methods for Itô equations, we propose to use a rather simple additional...

We consider a stochastic model of the two-dimensional chemostat as a diffusion process for the concentration of substrate and the concentration of biomass. The model allows for the washout phenomenon: the disappearance of the biomass inside the chemostat. We establish the Fokker–Planck...

Understanding the behaviour of market prices is not simple. Stock market prices tend to have complicated distributions with strong skewness and fat tails. One important step in forecasting tomorrow’s price is to estimate the volatility, i.e. how much tomorrow’s price is expected to differ...