In this paper, we investigate the dynamical behaviors of a Morris-Lecar neuron model. By using bifurcation methods and numerical simulations, we examine the global structure of bifurcations of the model. Results are summarized in various two-parameter bifurcation diagrams with the stimulating current as the abscissa and the other parameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay much attention to the emergence of periodic solutions and bistability. Different membrane excitability is obtained by bifurcation analysis and frequency-current curves. The alteration of the membrane properties of the Morris-Lecar neurons is discussed.

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh-Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov-Takens bifurcation which includes a saddle-node bifurcation, an Andronov-Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.

In this paper, a predator-prey model with disease in the prey is considered. Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. We study the dynamics of the model in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. We discuss the Bogdanov-Takens bifurcation near the boundary equilibrium and the Hopf bifurcation near the positive equilibrium; numerical simulation results are given to support the theoretical predictions.

No abstract

No abstract

No abstract

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der PolDuffing oscillator system is given.

No abstract