Periodicity and attractivity of a ratio-dependent Leslie system with impulses

A ratio-dependent Leslie system with impulses is studied. By using a comparison theorem, continuation theorem base on coincidence degree and constructing a suitable Lyapunov function, we establish sufficient and necessary conditions for the existence and global attractivity of periodic solution. Examples show that the obtained criteria are easily verifiable.     

Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0

In this paper, we consider the superquadratic second order Hamiltonian system

     

Recurrent points and non-wandering points of graph maps

Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps.     

Diagonalization of tensors with circulant structure

The concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors. It is shown a tensor of arbitrary order, which is circulant with respect to two particular modes, can be diagonalized in those modes by discrete Fourier transforms. This property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure. Tensors with circulant structure occur in models for image blurring with periodic boundary conditions. It is shown that the new framework can be applied to such problems.     

Exponential stability of periodic neural networks with impulsive effects and time-varying delays

By employing Young inequality and constructing suitable Liapunov functions, we investigate the existence and globally exponential stability of periodic neural networks with impulses and time-varying delays. The results extend and improve some earlier ones [1], [5] and [12]. An illustrative example and simulations are given to show the validity of the main results.     

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.     

Solitary wave and chaotic behavior of traveling wave solutions for the coupled KdV equations

Using the method of dynamical systems to study the coupled KdV system, some exact explicit parametric representations of the solitary wave and periodic wave solutions are obtained in the given parameter regions. Chaotic behavior of traveling wave solutions is determined.     

Positive periodic solutions of singular systems of first order ordinary differential equations

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

The periodic traveling-wave solutions of the short-pulse equation

In this paper, we consider the short-pulse equation (SPE) by using the dynamical system theory. The periodic solutions of the SPE are investigated. The solutions are characterized by two parameters. The periodic waves existent conditions are found, and explicit parametric expressions of all periodic solutions are given.     

Existence of two periodic solutions for a non-autonomous SIR epidemic model

A non-autonomous SIR model with periodic transmission rate and a constant removal rate is formulated. By using the continuation theorem of coincidence degree theory, sufficient conditions for the existence of at least two positive periodic solutions are obtained. The stability of the periodic solution for small seasonality is investigated. Numerical simulations are done to show our theoretical results.