Application of Extended Geometrical Criterion to Population Model with Two Time Delays

Geometrical criterion is a flexible method to be applied to a type of delay differential equations with delay dependent coefficient. The criterion is used to solve roots attribution of the related characteristic equation in complex plane effectively by introducing a new parameter skillfully. An extended geometrical criterion is developed to compute the stability of DDEs with two time delays. It is found that stability switching phenomena arise while equilibrium solution loses its stability and becomes unstable, then retrieve its stability again. Hopf bifurcation and the bifurcating periodic solution is analyzed by applying central manifold reduction method. The novel dynamical behaviors such as periodical solution bifurcating to chaos are discovered by using numerical simulation method.

BIFURCATION AND CHAOS ANALYSIS FOR A DELAYED TWO-NEURAL NETWORK WITH A VARIATION SLOPE RATIO IN THE ACTIVATION FUNCTION

In this paper, some complex phenomena of dynamical bifurcations are shown for a two-neural network with delay coupling. The sigmoid activation function with slope ratio, a monotonically increasing function, is proposed to consider the relations of the sigmoid and Hardlim functions. The equilibrium points are studied analytically in detail in terms of the characteristic equation and static bifurcation. The central manifold reduction and normal form method are employed to determine Hopf bifurcation and its stability. The stable equilibrium points and periodic motions are observed in different parameter regions. Effects of slope ratio and delayed coupling on dynamic behaviors are investigated by the numerical simulation, such as Poincare map, phase portraits, and power spectrum. Various active transitions to chaos and the corresponding critical boundaries on the focused parameter regions are obtained to classify dynamical behaviors including stable equilibrium point, periodic solution, 2-torus, 3-torus, and then the chaotic motions.