Dynamics Analysis On Neural Networks With Time Delays

In the past two decades, a large number of delayed neural networks have beenproposed to solve various engineering problems. The design for suchneural-network-based computational systems entails a profound understanding of thedynamics. In general, the study on dynamic behaviors in delayed neural networks is notonly of theoretical significant but also has important applications such as dynamicallyassociative memories and so on. Consequently, the dynamical issues of delayed neuralnetworks have been received considerable attention. Recently, many interesting resultssuch as global stability （including absolute stability, asymptotic stability, robust stability,and exponential stability, etc.） criteria for the equilibriums or periodic solutions,bifurcation and chaos of delayed neural networks have been derived.This dissertation focuses on stability and bifurcation analysis of ring structureneural network, Codimension-two bifurcation of BAM neural network and inertialneural network. The main contributions and originality contained in this dissertation areas follows:①Stability and bifurcation analysis in a class of ring structure neural networkwith time delaysRing networks have been found in a variety of neural structures such as neocortex,cerebellum and hippocampus, even in chemistry and electrical engineering. By studyingring networks, we can gain insight into the mechanisms underlying the behavior ofrecurrent networks. In this thesis, a class of n-dimensional BAM neural networks withring topology and time-varying delays is considered. Some necessary and sufficientconditions are derived for delay independently local stability of such neural network inthe parameter space. Then, the delay dependent stability and Hopf bifurcation areinvestigated. By analyzing the associated characteristic equation, it is found thatalthough such networks may have multiple delays and different connection strengthsamong individual nodes, the local stability and Hopf bifurcation are dependent on thesum of all time delays among all elements and the product of the connection strengthsbetween all elements. Moreover, the sufficient condition for the global exponentialstability of the ring Structured neural network is also presented. Finally, a tri-neuronnetwork is employed to illustrate the theorems developed in this study.②Hopf-Pitchfork bifurcation in a simplified BAM neural network model with multiple delaysBy studying the distribution of the eigenvalues of the associated characteristicequation, 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 carry out bifurcation analysis and numerical simulations showing thatthere exist a stable fixed point, a pair of stable fixed points, a stable periodic solution,and co-existence of a pair of stable periodic solution in the neighborhood of theHopf-Pitchfork critical point.③Bogdanov-Takens bifurcation in a tri-neuron BAM neural network model withmultiple delaysWe show that the connection topology of the network plays a fundamental role inclassifying the rich dynamics and bifurcation phenomena. And there are a wide range ofdifferent dynamical behaviors which can be produced by varying the coupling strength.By choosing the connected weights c₂₁ and c₃₁（the connection weights through theneurons from J-layer to I-layer） as bifurcation parameters, the critical values whereBogdanov–Takens bifurcation occurs are derived. Then, by computing the normal formsfor the system, the bifurcation diagrams are obtained. Furthermore, some interestingphenomena, such as saddle-node bifurcation, Pitchfork bifurcation, homoclinicbifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found bychoosing the different connection strengths. Some numerical simulations are given tosupport the analytic results；④Hopf-Pitchfork bifurcation in an inertial two-neuron system with time delayBy studying the distribution of the eigenvalues of the corresponding transcendentalcharacteristic equation of the linearization of this system, we derive the critical valueswhere Hopf-Pitchfork bifurcation occurs. Then, by computing the normal forms for thesystem, the bifurcation diagrams are obtained. Furthermore, we find some interestingphenomena, such as the coexistence of two asymptotically stable states, two stableperiodic orbits, and two attractive quasi-periodic motions, which are verified by boththeoretically and numerically.；...