A kind of delay neural network with n elements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. 1988 and delay differential equations due to Wu 1998 , and combining the normal form theory of functional...

We examine numerically the influence of intrapulse Raman scattering (IRS) on the stable stationary pulses in the presence of constant linear and nonlinear gain as well as spectral filtering. Numerical results show that the small change of the value of the parameter describing IRS leads to qualitatively different behavior of the evolution of pulse amplitudes. We prove that the strong dependence ...

We propose a method to control noise-induced motion, based on using delayed feedback in the form of the difference between the delayed and the current states of the system. The method is applied to two different types of systems, namely, a selfoscillator near Andronov-Hopf bifurcation and a threshold system. In both cases we demonstrate that by variation of time delay one can effectively contro...

In this paper, a delayed prototype model is studied. Regarding the delay as a bifurcation parameter, we prove that a sequence of Hopf bifurcations will occur at the positive equilibrium when the delay increases. Using the normal form method and center manifold theory, some explicit formulae are worked out for determining the stability and the direction of the bifurcated periodic solutions. Fina...

We investigate dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analysed and yield the transcrit...

In this paper we consider the numerical solution of delay diierential equations (DDEs) undergoing a Hopf bifurcation. The aim is to understand what will happen when simple standard numerical methods are used to obtain an approximate solution. We present three distinctive and complementary approaches to the analysis which together provide us with the result that # methods applied to a DDE will r...

Some delay independent and delay dependent conditions are derived for the global stability of the bidirectional associative memory neural networks with delayed self-feedback. Regarding the self-connection delay as the parameter to be varied, the linear stability and Hopf bifurcation analysis are carried out. An algorithm to determine the direction and stability of the Hopf bifurcations is also ...

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two...

We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of mul...

We study the local Hopf bifurcations of codimension one and two which occur in the Shimizu-Morioka system. This system is a simplified model proposed for studying the dynamics of the well known Lorenz system for large Rayleigh numbers. We present an analytic study and their bifurcation diagrams of these kinds of Hopf bifurcation, showing the qualitative changes in the dynamics of its solutions ...