A Kind Of Bifurcation Behavior Of Discontinuous Images

In this paper,we study the bifurcation behavior of a discontinuous map.It is a simplified model for a relaxation oscillator.First,under the control of single parameter,the bifurcation diagram of the system is obtained with the change of control parameter.It finds that the map present period-decrease sequence,period-adding sequence and chaotic phenomenon.The results show that in the periodic region,due to the the border collision bifurcation of the system,it generate the period-adding sequence.It concludes that there are five kinds of collision methods for this system.We define the rotation number of the periodic orbits,and it form a phase-locked devil’s staircase that is a multiple devil’s staircase with structural detuning.In the phase-locked devil’s staircase,the phenomenon that the same period has different rotation number is found,and the border collision of the system has certain rules.With the change of control parameters,the non-invertible region of the map gradually covered the gap,which leads to the disappearance of the gap,we observe the discontinuous pitchfork bifurcation phenomenon and the intermittencies phenomenon.Second,under the control of two parameters,the bifurcation diagram shows that the system has periodic and chaotic states.In this paper,the bifurcation mechanism of the map is studied,and we theoretically analyzed the stable parametric space of each period by the border collision condition and the linear stability analysis.This result basically is consistent with the results of our numerical simulation.The phenomenon of attractor coexistence was found and validated.