Research On Bifurcation And Collective Behavior For Several Classes Of Nonlinear Dynamical Systems

The application of bifurcation theory and synchronization theory in the ecologi-cal model,the neuron model and the wheelset.model is discussed in this dissertation.For the ecological model,we apply the forward Euler scheme to discrete a continuous predator-prey model with Allee effect and obtain the corresponding discrete model,for which three kinds of codimension one bifurcations are analyzed.For the neuron mod-el,bursting firing synchronization behavior and bifurcation mechanism of two chaotic Rulkov neurons with electrical coupling are investigated.For the wheelset model,b-ifurcations of the equilibrium and limit cycles are discussed with the help of Matcont software.The organization of this dissertation is as follows:In Chapter 1,we mainly introduce the main research content of nonlinear dynam-ical systems,and describe the research significance and development of the population ecological models.neuron dynamical models and wheelset models.In Chapter 2,some background knowledge of the discussed models is given and some conceptions,theorems and propositions associated with this dissertation are listed.In Chapter 3,a continuous predator-prey model with Allee effect is discretized by the forward Euler scheme to obtain a discrete model.Based on the center manifold theorem and the normal form theory,the normal forms of the fold bifurcation,the flip bifurcation and the Neimark-Sacker bifurcation are derived theoretically.Effect of the integral step size on bifurcation behavior is discussed,and similarities and differences between the continuous and the discrete dynamical model are compared.The bifurca-tion structure of the models with strong and weak Allee effect is taken into account.Dynamical behavior of two heterogeneous chaotic Rulkov neurons with electrical coupling is investigated in Chapter 4.Firstly,we analyze the ability of one neuron to change and affect the dynamical behavior of another under the different coupling strength.Secondly,the coupling conditions to attain in-phase and anti-phase bursting synchronization are discussed for three different combinations.Finally,the transition a-mong bursting synchronization,spiking synchronization and complete synchronization are probed.It is shown that two heterogeneous chaotic Rulkov neurons with bursting firing patterns can achieve spiking synchronization at the very strong coupling strength,while they can not achieve complete synchronization at any strong coupling strength.The model of two heterogeneous chaotic Rulkov neurons with electrical coupling is further investigated in Chapter 5.A flip and a Neimark-Sacker bifurcation for codi-mension 1 and a flip-Neimark-Sacker bifurcation for codimension 2 are analyzed.We derive the normal forms of all the bifurcations and analyze the local dependency of normal forms on the initial values and bifurcation points,and find a new kind of firing pattern—bounding alternately between two invariant cycles.When bifurcation param-eters are chosen far away from the flip-Neimark-Sacker bifurcation point,some new firing pattern appears,which is two symmetrical multiple heart-shaped cycles.Finally,we divide the parameter plane into several parts and obtain the distribution diagram of different firing patterns.The bifurcation behavior of a simple nonlinear wheelset model is investigated in Chapter 6.Wherein the Hopf and the generalized Hopf bifurcation of the equilibrium O are discussed,and the corresponding bifurcation curves are also given.As the continu-ation of limit cycles,the fold bifurcation,the pitchfork bifurcation,the flip bifurcation and the Neimark-Sacker bifurcation of limit cycles are analyzed,and the corresponding bifurcation curves of limit cycles with strong resonance points are exhibited.The effect of some nonlinear term on the bifurcation structure of limit cycles is discussed in the end.