The Dynamic Behaver Of 1D Discontinuous Piece Wise-linear Maps With The Same Intercept Sign

Piecewise-smooth systems can be classified into two main categories:continuous and discontinuous piecewise-linear systems.The research on the dynamic behaviors of continuous piecewise-smooth systems is quite mature,so in this article we focus on the discontinuous systems.As a part of piecewise-smooth maps,piecewise-linear maps are valuable in both theory and application.We choose the 1D piecewise-linear model with the same intercept sign and one discontinuous point as our research objective.First of all,we inturduce the background and significance of the piecewise-smooth systems,and some research results in this field.We aim at searching for the border collision bifurcation curves,according to the parameters we classify the problem into two cases.Then we use Leonov approach to analyze border collision bifurcation curves.We also present the periodic region that formed by border collision bifurcation curves and flip bifurcation curves.The result has shown that when the 1D discontinuous piecewise-linear maps have the same intercept sign,it has a rich periodic structure.Then we explore other dynamic properties of the maps.We point out the condition for critical homoclinic orbits and introduce the noncritical homoclinic orbits and chaos when the new critical homoclinic orbits appear.What’s more,the chaotic attractors are studied in this article,we found that the chaotic attractors change with the parameters of the maps,and they are illustrated by images.