Research On Dynamical Behaviors For Two Classes Of Neuron Systems

In this dissertation,we discuss chaos,bifurcations and synchronization of the neuron and the neuroendocrine systems.The following contents are mainly considered: first,the existence of chaos in the Rulkov neuron model and its one-dimensional fast subsystem is proved based on Marotto’s theorem.Second,the intermittent evolution routes to the asymptotic regimes in Rulkov map are introduced.That is,the windows with transient approximate periodic and transient chaotic behaviours occur alternatively before the system reaches the periodic or the chaotic orbits.Third,the existence conditions,asymptotic regimes and intermittent routes of synchronization manifold and complete synchronization of a three-layer Rulkov neuron network model are analyzed.Finally,the existence and properties of Hopf bifurcation,and the transient solution spaces of a three-state-variable Goodwin model with multiple time delays in Hill functions are investigated.The dissertation is summarized as follows:In chapters 1 and 2,we introduce the research background,research status and some basic concepts and theorems in nonlinear dynamical systems,especially in the field of chaos,respectively.In chapter 3,the existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem.First,the stability conditions of the model are briefly renewed.Second,for the two-dimensional Rulkov neuron model,it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces.Third,the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast-slow dynamics technique,in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods.Chapter 4 introduces the intermittent evolution routes to the asymptotic regimes in Rulkov map.That is,the windows with transient approximate periodic and transient chaotic behaviours occur alternatively before the system reaches the periodic or the chaotic orbits.Meanwhile,the evolution routes to chaotic orbits can be classified into different types according to the windows before reaching asymptotic chaotic states.The initial values can be regarded as a key factor to affect the asymptotic behaviours and the evolution routes.Finally,some experiments show that there is an about 60% accuracy rate of successfully predicting both the evolution routes and the asymptotic period-3 orbits using a three-layer feedforward neural networks,while the bifurcation diagrams can be reconstructed using reservoir computing except for a few parameter conditions.Chapter 5 concerns a three-layer Rulkov neuron network coupled by electrical synapses in the same layer and chemical synapses in the adjacent layers.The existence conditions and asymptotic regimes of the synchronization manifold are given.Whereafter,two kinds of evolution routes to the asymptotic behaviors over time are presented.We also analyze the stability of master stability equations for the network that has no zero eigenvalues in the outer matrix.The results indicate that the network with the periodic synchronization manifold almost can achieve complete synchronization,while the network with the chaotic synchronization manifold can not.The effects of small perturbations on the asymptotic regimes and evolution routes are also simulated.In chapter 6,the Hopf bifurcation of a three-state-variable delayed Goodwin model with multiple Hill functions is considered.The direction and the stability of Hopf bifurcation are also analyzed using the normal form theory and the center manifold theorem for functional differential equations.Furthermore,based on the sparse identification algorithm,it is verified that the transient time series generated from the delayed Goodwin model can not be equivalently presented by ordinary differential equations from the viewpoint of data when considering that a library of candidate are at most cubic terms.Finally,we report that reservoir computing can predict the periodic behaviors of the delayed Goodwin model accurately if the size of reservoir and the length of data used for training are large enough.