We mainly prove: Assume that each output function of DCNN is bounded on R and satisfies the Lipschitz condition, if is a periodic function with period ω each i, then DCNN has a unique ω-period solution and all other solutions of DCNN converge exponentially to it, where is a Lipschitz constant of for i=1,2,...,n.
In this paper, a discrete version of continuous non-autonomous predator-prey model with infected prey is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and global asymptotical stability of positive periodic solution of difference equations in consideration are established. An example shows the feasibility of the main results.
In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p = 2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the “superlinear” problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).
In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation xn+1 = axnxn−1 ⁄−bxn + cxn−2 , n ∈ N0 where a, b, c are positive real numbers and the initial conditions x−2, x−1, x0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with (a − c)/b < 1. When a > c with (a − c)/b > 1, we prove that every admissible solution is unbounded. Finally, when a = c, we prove that every admissible solution converges to zero.
A four-dimensional hyperchaotic Lü system with multiple time-delay controllers is considered in this paper. Based on the theory of Hopf bifurcation in delay system, we obtain a simple relationship between the parameters when the system has a periodic solution. Numerical simulations show that the assumption is a rational condition, choosing parameter in the determined region can control hyperchaotic Lü system well, the chaotic state is transformed to the periodic orbit. Finally, we consider the differences between the analysis of the hyperchaotic Lorenz system, hyperchaotic Chen system and hyperchaotic Lü system.
The problems related to periodic solutions of cellular neural networks (CNNs) involving D operator and proportional delays are considered. We shall present Topology degree theory and differential inequality technique for obtaining the existence of periodic solution to the considered neural networks. Furthermore, Laypunov functional method is used for studying global asymptotic stability of periodic solutions to the above system.
In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence xn+1 = a0xn + a1xn−1 + . . . + akxn−k ⁄ b0xn + b1xn−1 + . . . + bkxn−k , n = 0, 1, . . . where the parameters ai and bi for i = 0, 1, . . . , k are positive real numbers and the initial conditions x−k, x−k+1, . . . , x0 are arbitrary positive numbers.
New results are proved on the maximum number of isolated $T$-periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.
Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The derivation is based on elementary analysis. However, it is hoped that our easy fixed point theorems have potential applications in exploring stationary states of similar biological network models.