Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Abstract

In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis.

Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

“…This shows that the final trajectories of the Toda oscillators are asymptotically stable in the synchronous region, that is to say, there is a regular stable transmission between the oscillators, and the industrial productivity reaches the maximum. Thus far, the bifurcation phenomenon caused by coupled delay in nonlinear differential equations has been extensively studied, especially, the Hopf-zero bifurcation [1,2,4,5,18,[26][27][28][29], but no studies have been conducted on the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay as we know today. Therefore, it is of great significance to probe into the Hopf-zero bifurcation of system (1).…”

Dynamical analysis of a competition and cooperation system with multiple delays

“…Many complicated and large-scale systems in nature and society can be modelled as DDEs due to their flexibility and generality for representing virtually any natural and man-made structure. Research of the dynamical behaviour of DDEs has received much attention in interdisciplinary subjects, including natural sciences [3][4][5][6], engineering [7,8], life sciences [9] and others [10][11][12][13][14][15][16]. In recent decades, scientists have focused on the stability and bifurcation phenomena of the continuous-time autonomous predator-prey system with multiple delays (see, for example, [17][18][19][20][21][22]).…”