Huiping Zhang

In this paper, we study the bifurcations (the birth and the annihilation) of relaxation oscillations in a nonsmooth, slow–fast biological model with predator mutual interference. We find that the number of relaxation oscillations will change via the bifurcation events including canard explosion and two homoclinic bifurcations under variations of the control parameter. During this dynamical process, the number of positive equilibria changes and their positions on the critical curve move. Based on geometric singular perturbation theory, canard theory and entry–exit function, we capture this dynamical process completely in a continuous way. Depending on the values of parameters, the number of relaxation oscillations in this model may be zero, one and at most two. All the theoretical predictions are verified by numerical simulations.

This paper studies the bifurcation mechanisms and the number of relaxation oscillations in a singular perturbation Leslie-Gower model with Allee effect in prey and cubic type piecewise nonlinear functional response. Our approaches are based on slow-fast normal form, entry-exit function and geometric singular perturbation theory. Under two topologically different assumptions on the critical curve, we find that the organization centers on the generation of relaxation oscillations are significantly different. When the left fold is lower than the right one, relaxation oscillations are created or vanish via supercritical and subcritical canard explosions and homoclinic bifurcation. The maximal number of relaxation oscillations in this scenario is three. However, when the left fold is higher than the right one, relaxation oscillations are created or vanish via inverse canard explosions and super-explosion. The maximal number of relaxation oscillations in this scenario is only one left since the transcritical bifurcation point can not enter into the generation of relaxation oscillations after the unique equilibrium passing through the left fold. All the theoretical predictions are verified by numerical simulations.

In this study, we consider the multiplicity of solutions for a new class of $p$-Laplacian fractional differential equations involving $\psi$-Caputo fractional derivative with instantaneous and non-instantaneous impulses. Since the $p$-Laplacian operator, $\psi$-Caputo fractional derivative and impulsive effects exist at the same time, it is difficult to establish the variational structure of the considered problem. By virtue of critical point theorems and variational methods, we give some new criteria to guarantee that the problem has at least two weak solutions and infinitely many weak solutions. Some recent results are improved and extended.

In this paper, we consider a class of $ p $-Laplacian Kirchhoff-type fractional differential equations with instantaneous and non-instantaneous impulses. The existence of at least two distinct weak solutions and infinitely many weak solutions is obtained based on variational methods.

The p‐Laplacian fractional differential equations have been studied extensively because of their numerous applications in science and engineering. In this study, a class of p‐Laplacian fractional differential equations with instantaneous and noninstantaneous impulses is considered. The existence and multiplicity results of solutions are obtained by using variational methods and critical point theory. In addition, some examples are given to prove the feasibility of the proposed results.

In this study, we consider a Kirchhoff-type second-order impulsive differential system with the Dirichlet boundary condition and obtain the existence and multiplicity of solutions to the impulsive problem via variational methods.

Value of shear wave elastography with maximal elasticity in differentiating benign and malignant solid focal liver lesions