The Quasi-periodic Oscillation Of Coupled Generators

The coupled generators have been widely applied to radio-engineering,electronics,neuroscience and bio-chemistry.The coupled generators can produce a series of complex phenomena as their parameters change.Through numerical simulation and calculating Lyapunov exponents,A.P.Kuznetsov et al.[2019,Physica D]found many dynamical behaviors of this model,such as Hopf bifurcation,Neimark-Sacker bifurcation,invariant tori,etc.In this thesis,the existence of the quasiperiodic two-dimensional invariant torus of the coupled generators is analyzed by KAM theory.Taking the coupling coefficient Mc and the frequency detuning coefficient ? as parameters,we obtain the condition that the characteristic equation of the linearized system has two pairs of pure imaginary roots.We analyze the existence of the two-dimensional invariant torus of the averaged system by using the central manifold theorem and the averaging method,and obtain the parameter set ? of the existence of the two-dimensional invariant torus.Finally,it is proved by a KAM theorem that for most of the parameters in ?,the coupled generators also have a quasiperiodic two-dimensional invariant torus.This is consistent with the numerical simulation results of A.P.Kuznetsov et al.