Dynamic Analysis Of Piecewise-linear System And Chaos Control

The non-smooth dynamical system is a typical nonlinear system, while the movement form of the piecewise linear elastic impact is very common in practical production and mechanical systems. It has important theoretical and practical significance to the further research about the movement mechanism and the system global dynamical behavior of the piecewise linear elastic impact system. In this thesis, a single-degree-of freedom and a two-degree-of freedom piecewise-linear elastic impact system are studied and its dynamic behaviors are analyzed through theoretical analysis and numerical simulation method. The chaos phenomenon always exists in the non smooth dynamical systems, while in most cases, it has the bad effect on the stability of the system. So it has the vital significance to study its mechanism and control methods. The control theory of the OGY method, the Delay Feedback Method and the Radial Basis Function Neural Network are introduced with a single degree of freedom rigid collision, single degree of freedom piecewise linear elastic collision and typical Henon map as an example, which is proved to be effective through theoretical analysis and numerical simulation method. The main work of this paper includes the following contents:The first chapter is the introduction part. The research background, present situation and the main research method of non-smooth dynamics and chaos control are mainly introduced. The research direction of the paper is established.In the second chapter, the dynamic behavior of a single-degree-freedom and a two-degree-freedom piecewise-linear elastic collision system is studied. On the above two situations, the models are established respectively and the eigenvalues of the Jacobian matrix has been derived. The motion behavior of the system and process leading to the chaos have been simulated by the method of numerical simulation with excitation frequency ω for single bifurcation parameters and with excitation frequency ω, exciting force f two parameters for the bifurcation parameter. Finally the stability of the system is analyzed by the Lyapunov index spectrum.In the third chapter, the effectiveness of OGY control method of chaotic vibration control is mainly studied. Based on the main thoughts and principles of OGY control equation, taking single degree of freedom rigid collision system as an example, its validity and limitation are studied.In the fourth chapter, we use the time-delayed feedback control method in the chaotic system. The single-degree-freedom piecewise-linear elastic collision system is controlled by the time delayed feedback control method and the effect of control chart has been gotten by numerical simulation, which shows the feasibility and stability of this method.In the fifth chapter, it is mainly studied the radial basis function neural network(RBF) which is used to control chaotic system. Based on the principle of radial basis function neural network, the parameters of the network trained respectively by K-clustering, K- nearby and LMS method, the Henon mapping dynamic state is analyzed with the phase diagram and Poincare map. Finally the RBF neural network is adopted to control the mapping, which proves that the method can be used in the orbital periodical stability of the chaotic system.