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Research on Several Problems of Planar Piecewise-linear Dynamical Systems

Author: XuBin
Tutor: YangLiJun
School: Tsinghua University
Course: Mathematics
Keywords: piecewise-smooth homoclinic bifurcation Filippov system Poincare′map
CLC: O19
Type: Master's thesis
Year: 2013
Downloads: 7
Quote: 0
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Non-smooth systems consist of piecewise smooth systems, the Filippov systems andhybrid systems. After hundreds of years of study, there are plenty of results on smoothdynamical systems; but non-smooth systems have emerged only in recent decades, andthere remain many issues to be studied. Non-smooth systems have many applications,such as machinery, circuit, control systems, etc. Therefore, the non-smooth systems haveimportant theoretical significance and wide application prospects.In this dissertation, we mainly study the existence of homoclinic orbits and the ho-moclinic bifurcations in planar piecewise-linear system. We will first introduce severalexisting research results on piecewise smooth systems, including boundary equilibriumbifurcations, periodic orbits and Hopf bifurcations. The existence of homoclinic orbit inthis system can be divided into two cases, one is a system formed by a visible saddle pointsand a visible focus (or a center), the other one is a system formed by the coincidence of theoriginal point and two nodes with inverse stability.In view of the two cases, we provide thenecessary and sufcient conditions for the existence of homoclinic orbit, and then analyzethe homoclinic bifurcation. Finally, we provide the Poincare′map of periobic orbits inFilippov system, based on which we can do some further study on the existence, stabilityand isolation of periodic orbits.

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CLC: > Mathematical sciences and chemical > Mathematics > Dynamical systems theory
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