Stabilizing Hopfield neural networks via inhibitory self-connections

The theory of monotone dynamical systems is employed to establish some sufficient conditions for the global attractivity of the Hopfield neural networks with finite distributed delays. The results show that self-inhibitory connections can be used to stabilize a delayed network provided the diagonal delays corresponding to the inhibitory self-connections are small enough.     

A quadratic system with two parallel straight-line-isoclines

In this paper, a quadratic system with two parallel straight-line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqian et al., Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society, Providence, RI, 1986]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight-line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner–Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003].     

一类五次系统的全局分支

本文利用多参数扰动法并进行定性分析,对一类三次哈密尔顿系统进行五次扰动,得到了五个极限环.     

Global Bifurcation of a Perturbed Double-Homoclinic Loop

This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.     

On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

Bifurcation of limit cycles and the cusp of ordern

The author first investigates the limit cycles bifurcating from a center for general two dimensional systems, and then proves the conjecture that any unfolding of the cusp of ordern has at mostn−1 limit cycles. Supported by the Chinese National Natural Science Foundation.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

Limit cycles of quadratic systems

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our canonical system, we prove in a different way that a quadratic system has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.     

Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems

This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems.     

A discrete dynamical system as a model of a neural network with generalized Hebbian synapses

We study the dynamical behavior of a discrete time dynamical system which can serve as a model of a learning process. We determine fixed points of this system and basins of attraction of attracting points. This system was studied by Fernanda Botelho and James J. Jamison in [A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547] but authors used its continuous counterpart to describe basins of attraction.     

Global exponential stability of Cohen-Grossberg neural networks with variable delays

A class of generalized Cohen-Grossberg neural networks（CGNNs） with variable de- lays are investigated. By introducing a new type of Lyapunov functional and applying the homeomorphism theory and inequality technique, some new conditions axe derived ensuring the existence and uniqueness of the equilibrium point and its global exponential stability for CGNNs. These results obtained are independent of delays, develop the existent outcome in the earlier literature and are very easily checked in practice.     

Global exponential stability of a class of neural networks with delays

This paper is concerned with the stability of neural networks with time-varying delays. Under assumption that the nonlinear stimulate functions are Lipschitz continuous, by means of generalized Halanay inequalities, Dini＇s derivative and functional analysis techniques, several globally exponential stability criteria are established, which are only dependent on the parameters of the system.     

Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model

In this paper, we complete the global qualitative analysis of the well-known FitzHugh–Nagumo neuronal model. In particular, studying global limit cycle bifurcations and applying the Wintner–Perko termination principle for multiple limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.     

一类对称五次系统的极限环分支

对一类对称五次近Hamilton系统在五次对称摄动下产生的极限环数目进行了研究.通过多参数摄动理论和定性分析,得到这类对称摄动下的五次系统至少可以存在28个极限环.     

粒子的相互作用、极限环和相变

从各种相互作用的规范理论出发,讨论了规范场方程的某些新的解,并引入了势,然后探讨了它们与极限环、各种奇异点的关系,最后论述了这些结果可能具有的粒子性质和相变等物理意义．     

On Hopf bifurcations of piecewise planar Hamiltonian systems

In this paper we study the number of limit cycles appearing in Hopf bifurcations of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.     

ON THE LIMIT CYCLES OF PLANAR AUTONOMOUS SYSTEMS

91. IntroductionThe theory of ~ cycles is a very aCtive research field of qualitative theory of ordinary~nilal equationS. There have been many mathemsticians studying the nonetistence,~ence and piqueness of ~ cycles for plane systems, and most atteattiope were paidto some special formS (see [2-4, 610] and the references cited therein)' As we know, for thegeneral system on the Planei = p(z,g), b = Q(z,g), (1.1)where P, Q: RZ - R are continuouSly dmerentiable, there are some well-knoWn res…     

Global asymptotic stability of stochastic reaction-diffusion neural networks with time delays in the leakage terms

In this paper, a class of stochastic reaction-diffusion neural networks with time delays in the leakage terms is investigated. By using the Lyapunov functional method and linear matrix inequality (LMI) approach, sufficient conditions are derived to ensure the global asymptotic stability of an equilibrium point of the networks in the mean square. The results can be easily solved by MATLAB LMI toolbox. Finally, a numerical example is given to demonstrate the effectiveness and conservativeness of our theoretical results.     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

Bifurcation of a three-unit neural network

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork-Hopf bifurcation points, Hopf-Hopf bifurcation points, and some higher codimension bifurcation points.