The Study Of Dynamics Of Two Kinds Of Discontinuous Systems Based On The Flow Switchability Theory

Discontinuous phenomena are common in the real life, such as friction, collision,pulse, etc. But, with the deepening of the research of discontinuous dynamical systems,we are aware that the continuous dynamical system can be seen as a special case of discontinuous dynamical systems and the investigation of many problems of the above phenomena can be translated into researching the corresponding discontinuous dynamical systems. Thus, the study of discontinuous dynamical systems is particularly important.The impulse system as a special case of discontinuous dynamical systems, we natu-rally want to use the theory of discontinuous dynamical systems to solve the a classical problem in impulsive differential systems - the pulse phenomenon of the differential sys-tems. In addition, the investigation of discontinuous dynamical systems can solve many problems in our lives, so we want to use the theory of the flow switchability of discontin-uous dynamical systems to study a specific problem in life. Based on this, this paper is divided into two parts.In chapter one, we study the impulsive differential system with impulse at variable times where the vector field function F ? C(R+ ×?,R~n),?(?) R~n is an open set, X =(x1,x2,…,xn)T ? ?, ? ? C1(R~n,R+), H ? C1(R~n,R~n), I ?(?,R~n). Suppose that X ? ? implies X + I(X)? ?.We consider the system (1.2.1) as a discontinuous dynamical system consisting of two uniquely continuous subsystems, obtaining some sufficient conditions that guarantee the absence and presence of pulse phenomena, using the flow switchability theory of discontinuous dynamical systems. Especially, the case of the solution of the system sliding on the impulse surface is realistic and play an important role in the problems of synchronization, but this is beyond the classical definition of the solution of the impulse differential systems. Thus, the definition of this case is defined in this part, presenting some sufficient conditions of the presence of this phenomena. In these results obtained in this part, the constraint of the pulse function ?, especially its derivative, is reduced.In chapter two, we investigate the dynamics of a cambered impact oscillator under periodic excitation. According to the conservation law of momentum, impact law, circu-lar motion and the relationship between the horizontal moving component of the ball and the tangential movement, we establish the equations of the motion in different conditions.Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the necessary and sufficient con-ditions of the stick motions and grazing motions for the cambered impact oscillator and the results reflecting the nature of the cambered impact oscillator are obtained math-ematically using the flow switchability theory of the discontinuous dynamical systems,from which it can be seen that such oscillator has more complicated, characteristic and rich dynamical behaviors.