Research On Bifurcations For Several Classes Of Nonlinear Systems Under Random Disturbance

This dissertation aims at the bifurcations for several classes of nonlinear systems under random disturbance.Dealing with the different forms of random disturbance by means of various analysis techniques,the influences of stochastic parameters on the normal form of Hopf bifurcation,Pitchfork bifurcation,Flip bifurcation and Bautin bifurcation are discussed systematically,and further the corresponding numerical results are done as well.Firstly,a stochastic model for high-dimensional systems which undergo Hopf bifurcation is established,and a general method for the transformation of the model is described through the stochastic averaging process.Then,taking a jerky system as the research object,the codimension one and codimension two bifurcations and corresponding normal forms of the system without random perturbation are given.Furthermore,both additive and multiplicative noises are added to the jerky system when it undergoes Hopf bifurcation.After performing stochastic averaging using the above method,the stability of the equilibrium solution of the system and the parametric conditions for the occurrence of stochastic D bifurcation and stochastic P bifurcation are analyzed.Then the calculation framework of random normal form for Pitchfork bifurcation in high dimensional systems under random disturbance is constructed.Taking the Lorenz system as an example,the first step is to prove that the equilibrium solution of the system with nondiagonal multiplicative noises is stochastically stable,and simulate the properties of orbits under different noise intensities and initial conditions;the second step is to perturb the bifurcation parameter which induces the occurrence of Pitchfork bifurcation by small noise,describe the analysis idea of random normal form with double control parameters,and deduce the random normal form of Pitchfork bifurcation for the system.Thirdly,the normal form of Flip bifurcation under random disturbance is investigated by two cases.For codimension one case,a Neimark-Sacker bifurcation,a 1:2 resonance and a Fold-Flip bifurcation are discussed.It is found that the system undergoes homoclinic bifurcation and heteroclinic bifurcation near 1:2resonance point,and a Hopf bifurcation and a Cusp bifurcation near Fold-Flip bifurcation point.For codimension two case,the behaviors of system are similar to the codimension one case when random parameter is imposed on linear term,while system undergoes only a Flip bifurcation as random parameter is imposed on nonlinear term.Besides,numerical simulations are given to show the disparity caused by the random disturbance between the two cases.Finally,for the system which contains the normal form of codimension two Bautin bifurcation with additive noise,the difficulties brought by higher-order terms in proving the existence and uniqueness of the solution and solving the stationary probability density and Lyapunov exponent are overcome,synchronisation of system and stability of the random equilibrium are discussed,and the differences from the deterministic one are compared.In addition,a classical laser model with resonator is analyzed and simulated as an example.