Qualitative Theory For Several Classes Of Continuous And Discontinuous Differential Systems

This paper is devoted to qualitative theorem for several classes of continuous and discontinuous differential systems,which included the determination of centerfocus and Hopf bifurcation in continuous and discontinuous differential systems,the global phase portraits in continuous and discontinuous differential systems,center conditions and bifurcation of limit cycles in piecewise differential systems,local critical bifurcations in piecewise differential systems.It is organized as five chapters.In the first chapter,we recall the historical background and the present progress of qualitative theorem for continuous and discontinuous differential systems.The main work of this paper are concluded as well.In the second chapter,we introduce how to use the Poincar?e disk to describe the global phase portraits for planar differential systems.Using the Poincar?e compactification,we get the global phase portraits for a class of two parameters GrayScott model.In this chapter,we find that it generates singularity bifurcation,Hopf bifurcation,homoclinic bifurcation and heterclinic bifurcation when the parameters change in some critical values,i.e.Bogdanov-Takens bifurcation.In the third chapter,we further investigated the global phase portraits in discontinuous differential systems base on the above researches.We introduce how to use algebraic method to determine the number and corresponding position of the finite singular points,and how to use the index to judge the type of singularity.Using this method,we obtain the global phase portraits and bifurcation diagrams in a class of continuous and discontinuous Hamiltonian systems.In the fourth chapter,we introduce the method for studying the center-focus problem and Hopf bifurcation at the finite points of piecewise differential systems.Using this way,we obtain the center conditions and the number of limit cycles at the origin in a class of piecewise cubic systems.We present the way of the construction of the displacement function and the calculation method of Liapunov constants at the infinity of the piecewise differential systems.We further study center conditions and limit cycles at the infinity of these piecewise cubic systems.In addition,we study the number of limit cycles at infinity of another piecewise cubic differential systems.In the fifth chapter,we introduce the construction of the periodic function at the center,the calculation method of the periodic constant and the local bifurcation of critical periods in the piecewise differential systems.We study the bi-center conditions in a class of piecewise differential systems.By calculating the period constants,we obtain the number of local critical periods at the center(1,0)(or(-1,0))of these systems.