Symmetry And Bifurcation Of A Class Of Piecewise Smooth System Mapping

Non-smooth Vibration system is one of the more common systems in daily life.As a typical representative of nonlinear systems,it is of great significance to the research in related fields at home and abroad.The collision and impact phenomena will have an adverse impact on people's daily production and life.Therefore,studying the dynamic behavior of non-smooth systems is of great significance to daily life.In this paper,we mainly study the bifurcation characteristics of a class of piecewise smooth systems based on the nonlinear dynamic theory,and study the dynamic behaviors such as the stability of periodic orbits.The research object of this paper is a mechanical model of a two-degree-of-freedom symmetric vibration and shock system.First,the periodic solution of the system is obtained by analyzing the differential equations of the system dynamics,and the conditions that the periodic solution needs to meet are derived.Combined with the perturbation theory and other related knowledge,the periodic solution of the system under disturbed motion is obtained.Then the Poincaré map is constructed on the collision surface.According to the implicit function theorem,the linearized matrix of the system in the disturbed state is obtained.The numerical simulation is carried out with the help of C language,Grapher and other software to study the system under different excitation frequencies.The characteristics of periodic motion,and selected several sets of parameter values,obtained the periodic characteristics of system motion.The zero-time discontinuous map is constructed,and the dynamic behavior of the scrubbing motion is analyzed in conjunction with the relevant theory,and the singularity of the Jacobian matrix is obtained.Under the different parameters of the system's movement law,the numerical simulation method was used to draw the bifurcation diagram,phase diagram and time history diagram,and the stability of the system and the variation law of the bifurcation parameters were verified.