Researches On Bifurcation And Chaos Control Of Some Non-smooth Dynamical Systems

Non-smooth dynamical systems are abundant in both science and industrial fields.Due to the importance and complexity of non-smooth dynamical systems,more and more researchers begin to study them.In this thesis,three kinds of non-smooth dynamical systems are studied: a class of bi-nonlinear system with discontinuous coupling,a class of two-degree-of-freedom nonlinear conveyor system and a class of two-degree-of-freedom impact system with two-sided rigid constraints.Firstly,considering the complexity of dynamics caused by multiple constraints,a dynamic model with two discontinuously coupled bi-nonlinear oscillators and two non-smooth constraints is presented.The dynamic behaviors of the system with asymmetric clearances are investigated and rich bifurcation phenomena are revealed.Based on Four initial states of the asymmetric constraint system,for each initial state,the whole motion process of the oscillator is analyzed by path-following method.Numerical results indicate that there are many co-dimension-one bifurcations,for example,period-doubling bifurcation,Neimark-Sacker bifurcation and saddle-node bifurcation which can change the stability of the motions.Furthermore,three different co-dimension-two bifurcations,grazing-saddle-node bifurcation,saddle-node-Neimark-Sacker bifurcation and Neimark-Sacker-grazing bifurcation,are found through the intersections of grazing,saddle-node and Neimark-Sacker co-dimension-one bifurcation curves.Secondly,the model of two-degree-of-freedom nonlinear moving belt system is proposed,which considers the possibility of two-oscillator sticking or sliding on the moving belt simultaneously.By analyzing the overshooting region and sliding region of the two-oscillator switching manifold,the condition of the oscillators sticking on the moving belt is obtained.The periodic motions of the system are discussed by means of the numerical continuation based on the above regions and boundaries.The one-parameter bifurcation diagram of the sliding segment time and the overshooting segment time with respect to the velocity of the moving belt is presented respectively in order to understand the co-dimension-one sliding bifurcation of the system more intuitively.Then the two-parameter continuation of the co-dimension-one sliding bifurcation point is carried out to obtain the corresponding co-dimension-one bifurcation curve.Two types of co-dimension-two bifurcation points are obtained at the intersection of co-dimension-one bifurcation curves.Type-I co-dimension-two bifurcation points are the intersections of the co-dimension-one sliding bifurcation curves of one oscillator,and type-II co-dimension-two bifurcation points are the intersections of the co-dimension-one sliding bifurcation curves of different oscillators.The latter has rarely been presented in the literature.The unfolding of the dynamics of the system around the co-dimension-two bifurcation points is obtained.The numerical results show that the system exhibits rich and complex dynamic phenomena under the changes of conveyor belt speed and friction force.Finally,the model of two-degree-of-freedom impact system with two-sided rigid constraints is considered.In order to further study the dynamic behavior of the system,according to the global bifurcation diagram of the system,the periodic motion and chaotic phenomenon are alternately detected in a certain parameter range.The method of transformation between relative coordinates and absolute coordinates is introduced to solve the problem that the collision constraint surface is not fixed.And the Lyapunov exponent of two-sided rigid constraint system is calculated to verify the accuracy of the global bifurcation diagram.By adding damping coefficient and periodic excitation force,the chaotic motion of the system under the same parameter condition is controlled.The numerical results show that the chaotic phenomenon of the two-sided rigid constraint system can be effectively controlled to the periodic orbit.