Analysis Of The Boundary Oscillation Behavior Of Non-smooth Systems

Non-smooth systems are very common in real life,such as friction,collision systems,the systems of rectifying and inverting circuits and so on.There are discontinuous vector fields in non-smooth systems,and the boundary dynamic behavior of them has become a research hotspot and difficulty in nonlinear field.In this paper,the flow switching theory is used to analyze the three discontinuous systems,the motion states of the systems on the boundary are analyzed in detail,and the corresponding numerical simulation is given to verify the correctness of the theoretical analysis.The specific research contents are as follows:(1)For a nonsmooth Duffing oscillator with an absolute value term,the grazing and traversing motions of the flow passing through the displacement boundary are studied,and the analog circuit is completed.Based on the discontinuous dynamical theory,the motions of the non-smooth duffing system at the switching boundary is studied,and the corresponding analysis conditions of different motions are obtained,Through numerical simulations,chaotic motions and period orbits are described in detail with different parameters and initial conditions.The switching bifurcation diagrams through the boundary and basins of attractors are also drawn to investigate the behaviors of the system and coexistence of different attractors.(2)The nonlinear behaviors of a Duffing-like system with signum function are investigated through the theory of discontinuous dynamical systems,and the hardware circuit is realized based on FPGA digital technology.For a better understanding of the switching mechanism,the necessary and sufficient conditions of the Duffing-like system for motion switchability on the boundary are analyzed.The switching velocity with varying different system parameters and the parameter mappings are carried out to illustrate the dynamical motions.The attraction basins are depicted to express the coexistence of the Duffing-like oscillator with different initial values,and the coexisting trajectories in phase-space with various initial conditions are also exhibited.Through numerical simulations,the periodic and chaotic motions with different mapping structures are verified the effectiveness of the analysis conditions,respectively.Moreover,a hardware circuit of the Duffing-like system is established via Field Programmable Gate Array for the validation of the numerical analysis.(3)A high frequency current mode buck converter is analyzed through the flow switching theory,and the system simulation circuit is built to verify the accuracy of the analysis method.The motions of buck converter are analyzed,and the discontinuous motion states in the system is described in detail.The sufficient and necessary conditions for the system to change the motion state on the boundary are given by using the flow switching theory.The bifurcation diagrams of the system with reference current and clock pulse frequency is plotted to analyze the switching motions on the boundary.Based on Simulink platform,the simulation circuit is built to simulate the mapping structures,and the motion switch mechanism of the system on the boundary is analyzed in detail,which verifies the correctness of the theoretical analysis of the system.