Research On The Dynamics Of Several Nonlinear Systems

In this thesis,we study several nonlinear dynamic systems and explore their internal laws and mechanisms of some biological and physical phenomena,to predict their long-term evolution trends,and to formulate the required strategies.This thesis consists of the following three parts:The first part,Chapter 1,is an introduction to the dissertation.We present the development of nonlinear systems and summarize the main results of this thesis.The second part including Chapter 2,Chapter 3,Chapter 4,Chapter 5 and Chapter 6.In these five chapters,nonlinear systems with nonlocal delay or with periodic perturbation are studied specifically,and the main results can be briefly summed up as follows.In Chapter 2,we consider a diffusive predator-prey system with spatiotemporal delays.We study the local stability of the equilibria with the generalized delay kernels.Moreover,using the specific delay kernels,we perform a qualitative analysis of the solution,including the existence,uniqueness,and boundedness of the solution,global stability and Hopf bifurcation of the nontrivial equilibria.In Chapter 3,we study a diffusive nutrient-phytoplankton-zooplankton(NPZ)model with spatio-temporal delay.We advance the understanding of the local stability for equilibrium solutions of NPZ model by proposing a new local stability theorem for generalized three-dimensional systems.Using the specific delay kernel,we perform a qualitative analysis of the solutions,including existence,uniqueness,and boundedness of the solutions,global stability of the trivial equilibrium,and Hopf bifurcation of the nontrivial equilibrium.Numerical simulations are also performed to verify and supplement our analytical results.We show that diffusion predominantly has a stabilizing effect,however,if sufficient nutrient is present,complex spatio-temporal dynamics may occur.In Chapter 4,we propose a data-driven model with spatio-temporal delay to explore the transmission of COVID-19 during the strict border shutdown.The complete stability analysis of the COVID-19 model is given.On the basis of 96,233 laboratory-confirmed cases,the full-spectrum dynamics of COVID-19 in Canada and its regions are numerically reconstructed.Besides,as the long-term lockdown is socially and economically unsustainable,two shielding immunity strategies are implemented to reduce the epidemic peak and shorten the duration of epidemic spread.These results have important implications when considering continuing surveillance and interventions to contain outbreaks of COVID-19.In Chapter 5,we focus on bifurcations and related dynamical behaviors of a glucose metabolism model.It is shown that the model undergoes transcritical bifurcation,Hopf bifurcation,degenerate Hopf bifurcation,saddle-node bifurcation,cusp bifurcation and Bogdanov-Takens bifurcation of codimension 2 and codimension 3.Considering the periodicity of hepatic glucose production andβ cell’s glucose tolerance range,four elementary periodic mechanisms are also analyzed.These mechanisms lead to more complex dynamics,including periodic solutions of different periods,quasi-periodic solutions and chaos.Sensitive analysis and correlation matrix provide valuable insights to isolate the high-effect factors and to explore a few advanced treatment approaches.The described dynamics of this model well fit clinical glucose data and explain several experimental observations,which could provide good guidance in the therapeutic process.In Chapter 6,we investigate a repeated yielding model with or without periodic perturbation.For the unperturbed model,Hopf bifurcation,degenerate Hopf bifurcation,saddle-node bifurcation and zero-Hopf bifurcation are detected.For the periodically perturbed model,four periodic mechanisms are analyzed corresponding to five bifurcation cases of the unperturbed one.Rich dynamical behaviors arise,including stable and unstable periodic solutions of different periods,quasi-periodic solutions and chaos.According to the dynamics of the perturbed model,four types of stress-time curves and their trends are simulated,which can well interpret various experimental phenomena of repeated yielding.The last part is the conclusion and discussion.