Yancong Xu

Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function and CTL immune response

In this paper, we study a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence and CTL immune response. By constructing Lyapunov functionals, we show that if the basic reproductive number is less than or equal to one, then the infection-free equilibrium is globally asymptotically stable; if the immune reproductive number is less than or equal to one and the basic reproductive number is greater than one, then the immune-free equilibrium is globally asymptotically stable; if the immune reproductive number is greater than one, then the interior equilibrium is globally asymptotically stable.

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

Snakes and isolas in non-reversible conservative systems

Reversible variational partial differential equations such as the Swift–Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas.

A delayed virus infection model with cell-to-cell transmission and CTL immune response

In this paper, a delayed virus infection model with cell-to-cell transmission and CTL immune response is investigated. In the model, time delay is incorporated into the CTL response. By constructing Lyapunov functionals, global dynamical properties of the two boundary equilibria are established. Our results show that time delay in the CTL response process may lead to sustained oscillation. To further investigate the nature of the oscillation, we apply the method of multiple time scales to calculate the normal form on the center manifold of the model. At the end of the paper, numerical simulations are carried out, which support our theoretical results.

New Kamenev-type theorems for super-linear matrix differential equations☆

Abstract By using Riccati transformation and the integral averaging technique, some new Kamenev-type oscillation criteria are established for the super-linear matrix differential systems X ″ ( t ) + ( X n ( t ) Q ( t ) X ∗ n ( t ) ) X ( t ) = 0 and X ″ ( t ) + ( X ∗ n ( t ) Q ( t ) X n ( t ) ) X ( t ) = 0 , t ⩾ t 0 > 0 , n ≥ 1 , where Q ( t ) is an m × m continuous symmetric and positive definite matrix for t ∈ [ t 0 , ∞ ) . The results improve and complement those given by Tomastik [E.C. Tomastik, Oscillation of nonlinear matrix differential equations of second-order, Proc. Amer. Math. Soc. 19 (1968) 1427–1431], Ahlbrandt et al. [C.D. Ahlbrandt, J. Ridenhour, R.C. Thompson, Oscillation of super-linear matrix differential equation, Proc. Amer. Math. Soc. 105 (1989) 141–148] and Ou [L.M. Ou, Atkinson’s super-linear oscillation theorem for matrix dynamic equations on a time scale, J. Math. Anal. Appl. 299 (2004) 615–629], which is illustrated by an example at the end of the paper.