Differential Equations and Dynamical Systems

A set X together with a finite collection of self-maps on X is called a multiple discrete flow. We develop the theory of fair convergence for multiple flows and show how the UNITY programs of Chandy and Misra can be reformulated in terms of this theory.

In this paper we develop a statistic test in order to compare two discrete stochastic dynamical systems with autoregressive structures under heteroskedasticity, more specifically the variance of the errors follows a general function depending on time. The random disturbances or errors can also be contemporaneously correlated through some structure varying in time. The approach is based on a Wald test and we present three consistent estimates for the asymptotic covariance matrix. The study of the asymptotic properties is based on the L p -mixingale processes theory since we consider errors generated from martingale difference processes. In order to evaluate the performance of the test in finite samples, we ran some simulations with satisfactory results. A real application is also provided.

This paper addresses a stochastic differential game arising in a stock market largely controlled by big traders. We model stock price behaviour as a standard geometric Brownian motion and the stock market as characterized by the presence of a few large traders and a fringe of marginal “noise traders”. Using the concept of Nash equilibrium we compute the equilibrium strategies and optimal value functions for the large traders. We also establish the stability of the state process under equilibrium strategies of the large traders. Finally we illustrate our results through some numerical examples for each variation of our model.

This paper is concerned with the existence of multiple anti-periodic solutions to a nonlinear fourth order difference equation. The analysis is based on variational methods and critical point theory. Clark’s critical point theorem is used to prove the main results. An example illustrates the applicability of the results.

Strichartz estimates for a time-decaying harmonic oscillator were proven with some assumptions of coefficients for the time-decaying harmonic potentials. The main results of this paper are to remove these assumptions and to enable us to deal with the more general coefficient functions. Moreover, we also prove similar estimates for time-decaying homogeneous magnetic fields.

In this paper, we investigate the existence of periodic solutions of modified version of the Leslie-Gower predator-prey model with Holling-type II functional response in the presence of Michaelis-Menten type prey harvesting over a time scale. Sufficient conditions for the existence of periodic solutions are derived by using the continuation theorem of coincidence degree theory. The condition we obtain is easily verifiable and not much restricted.