On The Limit Cycles Of Two Kinds Of Nonlinear Oscillators

The problem of limit cycles is an important research topic in nonlinear dynamical systems,and its study is of great significance for a deeper understanding of the dynamic behaviors of the systems.This thesis mainly studies the problem of limit cycles of the Brusselator oscillator and the van der Pol oscillator.Through theoretical analysis and numerical simulations,the dynamic behavior of these two types of nonlinear oscillators is studied.This thesis is divided into four chapters.In chapter 1,we briefly introduce the research background and significance of this thesis,and give the main research work and innovation points.In chapter 2,the size and position of the limit cycle of the Brusselator oscillator are studied by using the renormalization group method.The research process is simplified by introducing parameter δ to transform the two-parameter research problem into the single parameter problem.The theoretical results obtained show that for sufficiently small ,the second-order perturbation can correct the errors existing in the first-order perturbation and give the better estimation of the size and position of the limit cycle.Numerical simulations verify the validity of the obtained theoretical results.In chapter 3,we mainly study the rhythmic dynamic behaviors of a tri-rhythmic van der Pol oscillator,that is,the change in the number of the limit cycles.Firstly,we use the energy averaging method to study the effect of time-delay feedback on the rhythmic dynamic behaviors of the tri-rhythmic van der Pol oscillator.Secondly,we use the stochastic averaging method to obtain the stationary probability density function,and derive that the time-delay feedback parameters , and noise intensity can cause stochastic P bifurcations in the stochastic system.Finally,the correctness of the theoretical analysis is verified by Monte Carlo numerical simulations.From the two-parameter bifurcation diagram of time-delay feedback,it can be seen that adjusting time-delay feedback parameters can make the system exhibit different rhythmic dynamic behaviors such as tri-rhythmic,bi-rhythmic and mono-rhythmic.This result provides a mathematical basis for achieving ideal rhythmic dynamic behaviors in practical applications.The conclusion and discussion for the work of this thesis are presented in chapter 4.