The Study Of Stability And Bifurcation Of Delayed Neural Networks And The Analysis Of Period Distribution Of Pseudorandom Number Generators

As a subfield of dynamical systems, neural networks exhibiting the abundant dynamical behaviors. Due to their applications in pattern recognition, signal processing and optimized calculation, the dynamical issue of neural networks have attracted many scholars for researching. This thesis is concerned with the stability and bifurcation for neural networks.Pseudorandom number generators have found comprehensive applications in such areas as Monte Carlo simulation, computer games and cryptography, the issue of arithmetic theories of pesudorandom number generators have attracted many scholars for studying. This thesis is concerned with the period distribution of a class of nonlinear pseudorandom number generators-inversive pseudorandom number generators.This thesis consists of six chapters.1. Global exponential stability of neural networks with mixed time delays on time scalesNeither the boundedness nor the monotony on the activation functions is assumed. By constructing Lyapunov-Krasovskii functional, employing the linear matrix inequality (LMI) approach, the homeomorphism map theory and the time scale theory, several sufficient conditions in LMI form are obtained to ensure the existence, uniqueness and global exponential stability of equilibrium point for the neural networks on time scales. The results obtained generalize the existing results.2. Boundedness and complete stability of complex-valued neural networks with Time DelayBy employing the local inhibition approach, some conditions to guarantee the boundedness of the complex-valued neural networks are obtained. Moreover, under the boundedness conditions, a compact set which globally attracts all the trajectories of the network is also given. In addition, several conditions in terms of real-valued linear matrix inequalities (LMIs) for complete stability of the complex-valued neural networks are established via the energy minimization method, Largrage mean value theorem and the approach which converts the complex-valued LMIs to the real-valued ones. It is worthy pointing out that this paper not only does not assume the equilibrium points to be isolated, but also generalize the existing results on boundedness and complete stability for real-valued neural networks to complex-valued neural networks.3. Stability and Hopf bifurcation analysis of a tri-neuron BAM neural network with distributed delayBy regarding the finite distributed as the bifurcating parameter, the dynamic behaviors in terms of local asymptotical stability and local Hopf bifurcation is studied. At the very beginning, by analyzing the root distribution of the characteristic polynomial, the critical values of the Hopf bifurcation is given. Moreover, the direction and stability of bifurcating periodic solutions are also derived by the normal form theory and the center manifold theorem. The results obtained generalize the existing results.4. Period distribution of inversive pseudorandom number generators over the Galois ringsBy employing the transformation method which converts the inversive pseudorandom number generators to the linear feedback shift registers, further, by reducing the period distribution of the linear feedback shift registers to the period distribution of the order of the roots of the characteristic polynomial. By analyzing, the full knowlegde of the period distribution of inversive pseudorandom number generators is obtained. The results obtained generalize the existing results.