On the periodicity of solutions of max‐type difference equation

This paper is devoted to study the periodic nature of the solution of the following max‐type difference equation: where the initial conditions x_?2,x_?1,x₀ are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

Extension on peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa‐Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirmthe qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

First‐order systems in on with absolutely continuous real symmetric π‐periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet‐type boundary value problems on [0,π] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet‐type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive periodic solution of a hepatitis B virus infection model

In this paper, a periodic model for hepatitis B virus infection is proposed. The model describes the breeding of the infected cells and the periodic variation of the environment. On the basis of the continuation theorem of coincidence degree theory, a condition for the existence of a positive periodic solution to the model is established. The result can be used to explain the wave phenomenon on the density of the pathogens in patients blood and the occurrence of superinfection in hepatitis B virus infections. Copyright © 2014 John Wiley & Sons, Ltd.     

A prey–predator interaction under fluctuating level water

We consider three species system, namely prey (roach), predator (pike) and plankton. We derive persistence and extinction conditions of the populations. Using coincidence degree theory, we determine conditions for which the system has a periodic solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Both discrete and distributed delays are considered in a two‐neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf‐pitchfork bifurcation. First, we obtain the codimension‐2 unfolding with original parameters for Hopf‐pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf‐pitchfork bifurcation with other Hopf‐fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons. Copyright © 2015 JohnWiley & Sons, Ltd.     

Bifurcation from single periodic orbit in discontinuous autonomous systems

This article is devoted to the study of bifurcations of periodic solutions for discontinuous autonomous systems from single periodic solutions of unperturbed discontinuous equations. In addition, local asymptotic properties of derived perturbed periodic solutions are also investigated. An example of planar discontinuous ordinary differential equations is given to illustrate the theory.