Hidden Attractors For Two Classes Of Nonlinear Autonomous Systems And The Research Of Hopf Bifurcation

The hidden attractor is a new attractor.Unlike the typical Lorenz,Rossler,and Chen attractors,it does not contain the neighborhood of the equilibrium point,nor can it use traditional calculation methods to calculate these hidden attractors.The first chapter recounts the background and significance of attractor theory research,and introduces some preliminary knowledge of hidden attractors,such as the idea of new analytic-l numerical algorithm,harmonic linearization and the positioning of stable periodic solutions.The second chapter takes a kind of nonlinear Van der Pol-Duffing oscillators as the research object,and studies the hidden attractors of this nonlinear system.Firstly,calculating the characteristic equations of the system and according to the Routh-Hurwitz criterion,then take the stability of equilibrium.Secondly,because the system has a pair of pure characteristic roots,using the Hopf bifurcation theory,it is concluded that the Hopf bifurcation will appear.Again,the original system is transformed by description,and a series of continuous functions are introduced.The analytic-numerical algorithm of the iterative sequence of the system is combined with the harmonic linearization method of the positioning stable periodic solution to locate the hidden attractor of original system.Finally,the numerical simulation is carried out by Matlab to obtain the specific phase diagram.The third chapter studies a kind of modified Chua system model.Taking ? the branch parameters as a branch parameter,the equilibrium characteristics are discussed,and the Hopf bifurcation theory is used to determine what conditions are improved in the modified Chua system.The analyticl-numerical algorithm is combined with the harmonic linearization method to locate the hidden attractor of original system.Finally,the theoretical analysis results are verified by Matlab numerical simulation,which is indeed present in this system.The fourth chapter mainly summarizes the two system models of the full-text research,and also looks forward to the hidden attractor problem of the later research.