Stability and Bifurcation in Delay–Differential Equations with Two Delays

Abstract

The purpose of this paper is to study a class of differential᎐difference equations with two delays. First, we investigate the local stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. General stability criteria involving the delays and the parameters are obtained. Second, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits the Hopf bifurcation. The stability of the bifurcating periodic solutions are determined by using the center manifold theorem and the normal form theory. Finally, as an example, we analyze a simple motor control equation with two delays. Our results improve some of the existing results on this equation. ᮊ

A continuous time Cournot duopoly with delays

“…In the particular case in which time delays in production are the same for both firms, we have also applied some recent techniques introduced by Chen et al (2013). In order to study the occurrence of Hopf bifurcations when time delays in production are different, a possible extension of the present work may be the use of analytical and geometrical results proposed by Lin and Wang (2012). The paper has also stressed the possibility of other dynamic phenomena, such as synchronisation failures and chaotic dynamics.…”

Time effect on the dynamical behavior of a life energy system dynamic model

“…From an ecological viewpoint, the interaction parameter areas of coexistence for two components are desirable. This method is important for understanding the regulatory chemical mechanisms of ecological systems, which provides a control mechanism to ensure a coexistence transition from an equilibrium to a periodic oscillation with desired amplitude and robust period [2,5,15,16,22,23,26]. In addition, we would like to point out that we can also investigate the Hopf bifurcation of system (1.3) by choosing the delay τ 2 as bifurcation parameter.…”

“…More advanced theory on energy [9][10][11][12][13][14][15][16][17][18][19][20] was put forward to endow the energy with profound connotation in a chemical and thermodynamic foundation. Over the past decades, energy system dynamic theory has attracted a great deal of attention and many excellent and interesting results have been reported.…”

“…For example, Kooijman [10] developed the dynamic energy budget theory, Kooijman et al [11] illustrated the application of the dynamic energy budget theory by analyzing some predatorprey systems, Kooijman et al [12] discussed the application of mass and energy conservation laws in physiologically structured population models of heterotrophic organisms, and Kooijman and Troost [13] explained how the dynamic energy budget theory's various modules. For more related works on this aspect, one can refer to [11][12][13][14][15][16][17][18][19].…”

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

“…Due to the complexity of multiple delays, the analysis of stability and Hopf bifurcation for the multi-delayed differential equations is far from complete. Many researchers tried to fill in some ''piece of the puzzle'' for the multiple delays problem (Li et al 1999). Considered the average/sum of time delays as the variable parameter and used the time-scale transformation, some multipledelayed neural network systems with simple architecture can be transformed to the equivalent models with one single delay (Cao and Xiao 2007;Mao and Hu 2008;Wei and Zhang 2008).…”