Bifurcations Of Several Periodically Forced Systems And Applications

The periodically forced differential equations are a special class of non-autonomous differential systems,where the right side of the equations is described by time-periodic functions.It has important theoretical significance and is widely used in many disciplines.In this thesis,using bifurcation theory of differential equations,we focus on several typically nonlinear dynamical systems with peri-odic forcing and study their dynamic behaviors including bifurcation of equilib-rium point,bifurcation of periodic solution and chaotic attractor.Based on the applicability and evolution of the model,we analyze the dynamic behaviors and provide some theoretical foundations for explaining and studying some nonlinear phenomena observed in the system.Firstly,the pumping effect in 1 pipe-1 tank and 1 pipe-2 tanks flow con-figurations is studied by periodically forced differential equations.We discover a novel characteristic that a nT-periodic solution(subharmonic solution,n? 2 and n ? Z)for the corresponding differential equations will lead to the pumping effect in the configurations with a T-periodic external excitation.Bifurcation the-ory reveals that nT-periodic solution can arise by period doubling bifurcations in the two models.In the 1 tank system,rich dynamical behaviors including Neimark-Sacker bifurcation,fold bifurcation,strong resonance points and coexis-tence of invariant circles will appear as parameters vary.For the 2 tanks system,a sufficient condition is given for the existence of the T-periodic solutions by us-ing topological degree theory.And we study the bifurcation of the T-periodic solutions.Then,a microbial continuous culture model with a periodically forced dilu-tion rate is discussed.Bifurcation diagrams for periodic solutions of the periodic forcing system are given by using numerical continuation method when the un-forced system undergoes supercritical and subcritical Hopf bifurcation.Stable and unstable quasi-periodic solutions,periodic solutions of various periods,can occur or disappear and even change their stability,when the periodically forced system undergoes Neimark-Sacker bifurcation,flip bifurcation,and fold bifurca-tion.Moreover,chaotic attractors will be generated by a cascade of period dou-blings.In addition,the various periodic solutions can well explain the oscillation phenomena observed in laboratory experiments.Next,a Kaldor-Kalecki model is considered to study the effect of interest rate on the phenomenon of business cycle.It is shown by the information from the People's Bank of China and the Federal Reserve System that interest rate is not a constant but with remarkable periodic volatility.Therefore,we consider periodically forced interest rate in the model and study its dynamics.It is found that,both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system,can lead to cyclical economic fluctuations.Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle,which agrees well with the pure monetary theory on business cycle.Moreover,the effect of periodic forcing is investigated on a system exhibiting a degenerate Hopf bifurcation.Two methods are employed to study the bifur-cations of the periodic solution in the periodically forced system.It is obtained by averaging method that the system undergoes fold bifurcation and even degen-erate Hopf bifurcation of periodic solution.On the other hand,it is also shown by Poincare map that the system will undergo fold bifurcation,transcritical bi-furcation,Neimark-Sacker bifurcation and flip bifurcation.Finally,we make a comparison between these two methods and corresponding results.