The bidirectional associative memory (BAM) neural network with four neurons and two delays is considered in the present paper.A linear stability analysis for the trivial equilibrium is firstly employed to provide a possible critical point at which a zero and a pair of pure imaginary eigenvalues occur in the corresponding characteristic equation.A fold-Hopf bifurcation is proved to happen at this critical point by the nonlinear analysis.The coupling strength and the delay are considered as bifurcation parameters to investigate the dynamical behaviors derived from the fold-Hopf bifurcation.Various dynamical behaviours are qualitatively classified in the neighbourhood of the fold-Hopf bifurcation point by using the center manifold reduction (CMR) together with the normal form.The bifurcating periodic solutions are expressed analytically in an approximate form.The validity of the results is shown by their consistency with the numerical simulation.
GE JuHong & XU Jian School of Aerospace Engineering and Applied Mechanics,Tongji University,Shanghai 200092,China
A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. The method is summarized as three steps, namely linear analysis at critical value, perturba- tion and increment for continuation. The PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. Meanwhile, the PIS not only inherits the advantages of the method of multiple scales (MMS) but also overcomes the disadvantages of the incremental harmonic balance (IHB) method. Three delayed systems are used as illustrative examples to demonstrate the validity of the present method. The periodic solution derived from the delay-induced Hopf bifurcation is obtained in a closed form by the PIS procedure. The validity of the results is shown by their consis- tency with the numerical simulation. Furthermore, an approximate solution can be calculated in any required accuracy.
