Sliding Bifurcation Of Piecewise Smooth Systems And Limit Cycles Of Higher Dimensional Systems

In this thesis, we shall study the sliding bifurcation phenomena of some classes of planar piecewise smooth systems with parameters. By using Filippov theory of piecewise smooth systems and qualita-tive theory of ODEs, we systematically study the sliding bifurcation and limit cycles of two classes of piecewise smooth linear-quadratic systems, we obtain some new bifurcation phenomena.Moreover, we apply the averaging methods and the Bezout's Theorem in Algebraic Geometry to prove that the number of limit cycles of polynomial differential systems in higher dimensional space is not only an exponential function in the dimension of the systems, but also a power function in the degree of the systems. Our results have extended and improved the known results.In the first part of the thesis, we shall study a class of planar piecewise smooth linear-quadratic differential systems with parame-ters. We prove that such systems can present very complicated and richer bifurcation phenomena, such as the sliding-crossing bifurca-tion, the sliding homoclinic bifurcation, the sliding Hopf bifurcation, the multiplicity 2 limit cycle bifurcation and so on. We also prove that this class of systems can have exactly two hyperbolic limit cycles. In addition, we study the case without sliding phenomena, where we obtain some results about the existence of heteroclinic and homo-clinic cycles and the existence and uniqueness of limit cycles. These results have extended and improved the previous known results.In the literature [Int. J. Bifur. Chaos.13 (2003),2157-2188], Kuznetsov and his co-authors analyzed the sliding-crossing bifurca-tion and sliding homoclinic bifurcation phenomena, they showed the existence of these phenomena via numercial analysis. We have proved the existence of these complicated bifurcation phenomena theoreti-cally via concrete systems. Moreover, we can obtain all these com-plicate bifurcation phenomena just through one system with param-eters.In the second part of this thesis, we shall analyze the sliding bifurcation through another class of planar piecewise smooth linear-quadratic systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. To our knowledge, the sliding heteroclinic bi-furcation is the new phenomenon that we first find. Furthermore, we will prove that the systems can present many bifurcation phenom-ena, such as the heteroclinic bifurcation, sliding homoclinic bifurca-tion and semistable limit cycle bifurcation and so on. This class of systems can have exactly two hyperbolic limit cycles.In the proof of our main results, we shall use the geometric singular perturbation theory to analyze the dynamics near the sliding region. This idea comes from the work given in the literature [C.A. Buzzi, P.R. da Silva and M.A. Teixeira, J. Diff. Eqns.231 (2006), 633-655]. However, there are some mistakes appearing in the choice of the blow-up intervals and phase portraits. We have corrected these mistakes and used the corrected results to prove the existence of the sliding heteroclinic bifurcation.In the third part of this thesis, we shall apply the averaging methods and the Bezout theory to study the Hopf bifurcation of a class of smooth differential systems in IRn. We prove that the number of limit cycles bifurcating from one singularity of the higher dimen-sional systems is not only an exponential function in the dimension of the systems, but also a power function in the degree of the sys-tems. These results have improved the ones given in in the literature [Llibre and Zhang, Pacific J. Math.240 (2009),321-341]. Their re-sults only proved that the number of limit cycles generated from the Hopf bifurcation was an exponential function in the dimension of the systems.In addition, we apply the averaging methods to study the limit cycles bifurcating from the periodic orbits of a class of four dimen- sional systems. If the perturbation function is a piecewise smooth degree n function, we prove that at most 3 limit cycles can bifurcate from the periodic orbits of this class of systems, where n≥2, n∈N. Our results have extended the ones given in the literature [C.A. Buzzi, J. Llibre and J.C. Medrado, On the limit cycles of a class of piecewise linear differential systems in IR4 with two zones, preprint,2010]. Their results only prove that if the pertuabation function was a piecewise smooth linear function, at most 3 limit cycles could bifurcate from the periodic orbits of this class of systems.