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

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.

[1]  L. Chua,et al.  Normal forms for nonlinear vector fields. I. Theory and algorithm , 1988 .

[2]  Chuandong Li,et al.  Bogdanov-Takens bifurcation in a single inertial neuron model with delay , 2012, Neurocomputing.

[3]  Zhigang Zeng,et al.  Pattern memory analysis based on stability theory of cellular neural networks , 2008 .

[4]  Zhigang Zeng,et al.  Global Stability of a General Class of Discrete-Time Recurrent Neural Networks , 2005, Neural Processing Letters.

[5]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[6]  J. Farrell,et al.  Qualitative analysis of neural networks , 1988, 1988., IEEE International Symposium on Circuits and Systems.

[7]  Junjie Wei,et al.  Bifurcation analysis in van der Pol's oscillator with delayed feedback , 2008 .

[8]  Junjie Wei,et al.  Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays , 2005 .

[9]  Chuandong Li,et al.  Fold-flip bifurcation analysis on a class of discrete-time neural network , 2011, Neural Computing and Applications.

[10]  L. Chua,et al.  Normal forms for nonlinear vector fields. II. Applications , 1989 .

[11]  Weihua Jiang,et al.  Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback , 2010 .

[12]  Zhigang Zeng,et al.  Design and Analysis of High-Capacity Associative Memories Based on a Class of Discrete-Time Recurrent Neural Networks , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[13]  Qiankun Song,et al.  LMI criteria on exponential stability of BAM neural networks with both time-varying delays and general activation functions , 2010, Math. Comput. Simul..

[14]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[15]  Liancheng Wang,et al.  Multi-parameter bifurcations of the Kaldor–Kalecki model of business cycles with delay , 2010 .

[16]  X. Liao,et al.  Dynamics of an inertial two-neuron system with time delay , 2009 .

[17]  X. Liao,et al.  Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus , 2012 .

[18]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[19]  Xiang-Ping Yan,et al.  Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays ☆ , 2008 .

[20]  Simon Haykin,et al.  Neural Networks: A Comprehensive Foundation , 1998 .

[21]  Fátima Drubi,et al.  Hopf-pitchfork singularities in coupled systems , 2011 .

[22]  C. Leung,et al.  Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function , 2001 .

[23]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[24]  Qiankun Song,et al.  Stability and Hopf bifurcation analysis of a tri-neuron BAM neural network with distributed delay , 2012, Neurocomputing.

[25]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[26]  S. Chow,et al.  Normal Forms and Bifurcation of Planar Vector Fields , 1994 .