Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Abstract

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self-and neighborconnection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the NeimarkSacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

Photosynthesis rate, salinity induced grazing rate and gestation delay as determinants of the dynamics of a realistic food chain model: a stochastic approach

“…It often represents gestation time (Jana 2014;Jana et al 2014), incubation period, transport delay or can simply lump complicated biological process together, accounting only for the time requires for these process to occur. Delay parameter emphasizes on Hopf-bifurcation (Ruan 2009;Jana 2014;Jana et al 2014Jana et al , 2015 and also it causes some complex dynamical behaviors, such as double Hopf bifurcation, quasi-periodic solutions and chaos (Lü 2002;Ma et al 2008;Matouk 2008;Ding and Jiang 2011;Wang et al 2014;Ding et al 2013;Song and Xu 2013). Along with different studies of environmental stochasticity and biological delays in prey-predator system, influence of noise on a delayed system is very significant for advance study of biological system.…”

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

“…Time delays occur naturally in many areas of physics, biology, chemistry, medicine, social science, and technology, such as nonlinear optics, electronic circuits, neuroscience, communication networks, and traffic flows [1,8,[26][27][28][29][30]. Time delays arise due to finite signal propagation and processing speeds, or may be introduced through closed-loop feedback control schemes.…”

Double Hopf Bifurcation for an Hopfield Eural Network Model With Time Delayed Feedback

“…In order to reach a deep and clear understanding of the dynamics of such models, most researchers have limited their study to the of models with a single delay [5,11]. In some papers, multiple delays are considered but there are no self-connection terms and moreover the systems with two delays have been generally investigated [8][9]. For example, Song, and Xu [8], investigated the stability of a two-neuron system with different time delay as follows:…”

“…In some papers, multiple delays are considered but there are no self-connection terms and moreover the systems with two delays have been generally investigated [8][9]. For example, Song, and Xu [8], investigated the stability of a two-neuron system with different time delay as follows:…”