Hopf-pitchfork bifurcation in an inertial two-neuron system with time delay

Authors:
Tao Dong;Xiaofeng Liao;Tingwen Huang;Huaqing Li
Affiliations:
State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 400044, China and College of Software and Engineering, Chongq ...;State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 400044, China;Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatar;State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 400044, China
Venue:
Neurocomputing
Year:
2012

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Abstract

In this paper, we have considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By studying the distribution of the eigenvalues of the corresponding transcendental characteristic equation of the linearization of this equation, we derive the critical values where Hopf-pitchfork bifurcation occurs. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically.