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Equilibria Property of a Three-parameter Piecewise Linear Planar Filippov Equation

Author: GuoYing
Tutor: ZouYongKui
School: Jilin University
Course: Computational Mathematics
Keywords: Volterra Filippov equation Balance Point Singular point Discontinuous Line Piecewise linear
CLC: O241.82
Type: Master's thesis
Year: 2010
Downloads: 12
Quote: 0
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Abstract

Ordinary Differential Equations is a equation that contain a independent variable and its unknown function and the differential of the unknown function,Differential Equations almost produced with Calculus,Its formation and development are closely related to mechanics, astronomy, physics and other science and technology development.Therefore, the ordinary differential equations is increasingly becoming a hot issue. With the detailed study,Piecewise smooth ordinary differential equations get more attention by the experts and scholars,In real life, piecewise smooth Ordinary Differential Equations’s Applications is very broad,Such as automatic control, set the field of electronics devices,Trajectory calculation, aircraft and mis-sile flight stability,Seismic systems, control theory, feedback systems, etc..These problems can be transformed for the sake of smooth or piecewise smooth differential equation,Or into seeking smooth or the nature of piecewise smooth differential equations.there are many similar places in the nature of the branch of Piecewise smooth ordinary differential equations and smooth ordinary differential equations.Relative to smooth equation , the difference is that it not only has all the phenomenon of branches of smooth systems,But also because of its own nature of not smooth, it also have a special phenomenon that the smooth system does not have.And the phenomenon of these branches is more complicated than the smooth system,and its applications are more widely.the branch nature of Piecewise smooth equations plays an important role in the study of equation’s global structure and qualitative theoryAs smooth ordinary differential equations and piecewise smooth ordinary differential equations both have a stable or unstable equilibrium point,In real problems, according to the change of the equation’s given parameters,generally we should understand the changes of the balance equation,When changes in the given parameters tends to a critical value,If the num-ber of balance point or stability are changed, then we call this phenomenon is equilibrium branch.the phenomenon of equilibrium branch is the hot topics of ordinary differential equa-tions,for a single parameter or two-parameter piecewise linear planar Filippov-type equation has been made a lot of results, the paper plane on the study of the nature of equilibrium point branch of a three-parameter piecewise linear equations.Zang Lin make a study for a class of piecewise linear three-parameter planar Filippov differential equations branches of the nature of the equilibrium point in paper in a class of Filippov-type differential equations and the generalized Hopf bifurcation nature, this paper constructs a concrete example of his theory To explain its contents. Par This paper focus on the nature of equilibrium point branch of A 3-parameter plane piecewise linear Filippov equations,Specifically includes the following three aspects:par First:introduce the required background knowledge of the research questions,give the classical theory of ordinary differential equations Existence and uniqueness and Filippov sense solutions and the definition of equilibrium point and singular par Second:introduce the some basic properties of the three parameters piecewise linear planar Filippov equation and analysis the control action of the three parameters on the equilibrium points. Par Third:research the branch nature of equilibrium point of 3 parameter plane piecewise linear Filippov equation. the first we use of differential contains Theory point that with positive development of the time ,the existence of the solution of the initial value problem of this equation, further study of the unique and multiple solutions, and then given intermittent nature of online with sliding equation satisfied by the solution, Further analysis character of the balance of the equation on the branch line break,so we get the conclusion:(1)ifλ1=λ2=λ3=0, then there exist a unique equilibrium point on the discontinuous line of this equation, it’s O(0,0).(2)ifλ1=λ3=0,andλ1≠0,then all the points in M is equilibrium point.(3)ifλ2λ3≥0,|λ2|+|λ3|≠0,equation (3.2)has a unique equilibrium point on the discontinuous line M. among(4)ifλ2λ3<0, equation (3.2)has no equilibrium point on the discontinuous line M。

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