一类Filippov-型微分方程的分支性质及广义Hopf分支
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摘要
本文主要研究了一类Filippov-型常微分方程的平衡点、周期解分支和广义Hopf分支性质.我们首先研究了一类3-参数平面分片线性Filippov-型方程的平衡点分支,利用微分包含理论研究了当时间正向发展时初值问题的解的存在性及唯一性或多解性;利用Filippov凸方法给出了间断线上具有滑动性质的解所满足的微分方程,通过研究其性质证明了此类方程在间断线上存在平衡点的充要条件.之后,研究了一类具有标准形式的3-参数平面分片线性Filippov-型方程的周期解分支,构造了此类方程在间断线上的Poincaré映射,通过研究其非平凡不动点的性质得到了此类方程不具有滑动性质的周期解的存在性和稳定性;同时通过研究相平面中解轨道的性质得到了此类方程具有滑动性质的周期解的存在性以及同宿轨道和异宿轨道的存在性.最后研究了一类单参数3-维分片非线性Filippov-型方程发生广义Hopf分支的机制,分析了其分片线性近似方程周期解的存在性并提出了此类方程的分支函数,构造了此类非线性方程在间断面上的Poincaré映射,通过研究其非平凡不动点的性质获得了产生广义Hopf分支的充分条件,因而证明了分片光滑周期解的存在性.
In recent years,piecewise smooth systems are extensively applied to many aspects on physical systems and technological devices in engineering and applied science.Some physical systems may contain different modes,and the transition from one mode to another can sometimes be idealized as an instantaneous transition,since the transition time from one mode to another is usually extremely short comparing to the time scale for the development of the state in each individual mode,which is described by an ordinary differential equation.Thus the mathematical models of these physical systems can be modeled to piecewise smooth dynamical systems.Piecewise smooth models often appear in many different disciplines,such as power electronic converters[14,13,48,5],mechanical systems with clearances or elastic constraints[54,52,62,63],earthquake engineering [6,33,35,34],stick-slip models with frictions[4,30,51,38]and many other physical processes.
The bifurcation theory in smooth dynamical systems has been deeply studied.In particular,one cannot always apply these results to piecewise smooth systems due to their discontinuity.Piecewise smooth systems can exhibit all kinds of bifurcations which appear in smooth systems,but some bifurcation phenomena could behave quite differently from those in smooth systems due to the influence of the discontinuity of the system. The local bifurcations such as saddle-node bifurcation.transcritical bifurcation and pitchfork bifurcation present special features in piecewise smooth systems comparing to the counterparts in smooth systems.Especially the geometrical property of the discontinuity is involved in the study of the generalized Hopf bifurcation which yields the piecewise smooth bifurcating periodic orbits.Some bifurcations related to discontinuity which can not be seen in smooth systems,have been widely studied in piecewise smooth systems. such as border-collision[15,17],corner-collision[15],grazing bifurcation[7,13,14,36,37], discontinuous bifurcation[18,42],sliding bifurcation[19,38,39],non-smooth bifurcation [42,45]and multiple crossing bifurcation[43]and so on.
The bifurcations of stationary points and periodic solutions are always pop problems in dynamical systems.We are interested in the Filippov-type ordinary differential equations conposed by two sub-linear-systems.If both stationary points of the sub-systems lie on the line of discontinuity and concide with each other,the degree of structural instability of this stationary point,on the line of discontinuity is at least 3.Therefore we need at least three parameters to study the bifurcation properties of this Filippov-type equation, which becomes part of the work in this thesis.The generalized Hopf bifurcation in 2 -dimensional nonlinear piecewise smooth systems has been investigated in many papers [60,61,62,63].But there is few results about the generalized Hopf bifurcation in 3 -dimensional nonlinear piecewise systems,which becomes an important part of this thesis.
In this paper,we focus ourselves on studying stationary points,periodic solutions. homoclinic and heteoclinic bifurcations in Filippov-type systems.Our main work consists of:
●We study the existence of solutions as time increasing of a 3 -parameter piecewise linear Filippov-type system based upon the theory of differential inclusion.We also studied the multiple existence and uniqueness of solutions,and the solutions with sliding mode by analyzing the properties of the vector field defined on the line of discontinuity. Finaly,we investigate the bifurcation properties of stationary solutions of this 3 -parameter piecewise linear Filippov-type system.
●We study the bifurcation properties of periodic solutions with and without sliding motion,and the homoclinic and heteroclinic solutions in a 3 -parameter piecewise linear Filippov-type system.We construct the Poincarémap of each sub linear system on the line of discontinuity,in terms of which,we define a return map on line of discontinuity for the whole Filippov-type system.By searching the nontrivial fixed point of the return map we obtain the existence of piecewise smooth periodic solution without sliding motion.We also prove the existence of periodic solution with sliding motion and a special type homoor heteroclinic orbit,which may reaches the stationary point within finite time.
●We study the generalized Hopf bifurcation mechanism for a 1 -parameter 3 -di-minsional nonlinear Filippov-type ordinary differential equation.We first analyze the existence of periodic solutions of the piecewise linear approaching system.Then we investigate the Poincarémap of each sub-smooth-system,which leads to the return map of the whole system.We prove that the interaction of the eigen-structure of each subsystem and the discontinuity determines the generalized Hopf bifurcation phenomena.
In recent years,piecewise smooth systems are extensively applied to many aspects on physical systems and technological devices in engineering and applied science.Some physical systems may contain different modes,and the transition from one mode to another can sometimes be idealized as an instantaneous transition,since the transition time from one mode to another is usually extremely short comparing to the time scale for the development of the state in each individual mode,which is described by an ordinary differential equation.Thus the mathematical models of these physical systems can be modeled to piecewise smooth dynamical systems.Piecewise smooth models often appear in many different disciplines,such as power electronic converters[14,13,48,5],mechanical systems with clearances or elastic constraints[54,52,62,63],earthquake engineering [6,33,35,34],stick-slip models with frictions[4,30,51,38]and many other physical processes.
The bifurcation theory in smooth dynamical systems has been deeply studied.In particular,one cannot always apply these results to piecewise smooth systems due to their discontinuity.Piecewise smooth systems can exhibit all kinds of bifurcations which appear in smooth systems,but some bifurcation phenomena could behave quite differently from those in smooth systems due to the influence of the discontinuity of the system. The local bifurcations such as saddle-node bifurcation.transcritical bifurcation and pitchfork bifurcation present special features in piecewise smooth systems comparing to the counterparts in smooth systems.Especially the geometrical property of the discontinuity is involved in the study of the generalized Hopf bifurcation which yields the piecewise smooth bifurcating periodic orbits.Some bifurcations related to discontinuity which can not be seen in smooth systems,have been widely studied in piecewise smooth systems. such as border-collision[15,17],corner-collision[15],grazing bifurcation[7,13,14,36,37], discontinuous bifurcation[18,42],sliding bifurcation[19,38,39],non-smooth bifurcation [42,45]and multiple crossing bifurcation[43]and so on.
The bifurcations of stationary points and periodic solutions are always pop problems in dynamical systems.We are interested in the Filippov-type ordinary differential equations conposed by two sub-linear-systems.If both stationary points of the sub-systems lie on the line of discontinuity and concide with each other,the degree of structural instability of this stationary point,on the line of discontinuity is at least 3.Therefore we need at least three parameters to study the bifurcation properties of this Filippov-type equation, which becomes part of the work in this thesis.The generalized Hopf bifurcation in 2 -dimensional nonlinear piecewise smooth systems has been investigated in many papers [60,61,62,63].But there is few results about the generalized Hopf bifurcation in 3 -dimensional nonlinear piecewise systems,which becomes an important part of this thesis.
In this paper,we focus ourselves on studying stationary points,periodic solutions. homoclinic and heteoclinic bifurcations in Filippov-type systems.Our main work consists of:
●We study the existence of solutions as time increasing of a 3 -parameter piecewise linear Filippov-type system based upon the theory of differential inclusion.We also studied the multiple existence and uniqueness of solutions,and the solutions with sliding mode by analyzing the properties of the vector field defined on the line of discontinuity. Finaly,we investigate the bifurcation properties of stationary solutions of this 3 -parameter piecewise linear Filippov-type system.
●We study the bifurcation properties of periodic solutions with and without sliding motion,and the homoclinic and heteroclinic solutions in a 3 -parameter piecewise linear Filippov-type system.We construct the Poincarémap of each sub linear system on the line of discontinuity,in terms of which,we define a return map on line of discontinuity for the whole Filippov-type system.By searching the nontrivial fixed point of the return map we obtain the existence of piecewise smooth periodic solution without sliding motion.We also prove the existence of periodic solution with sliding motion and a special type homoor heteroclinic orbit,which may reaches the stationary point within finite time.
●We study the generalized Hopf bifurcation mechanism for a 1 -parameter 3 -di-minsional nonlinear Filippov-type ordinary differential equation.We first analyze the existence of periodic solutions of the piecewise linear approaching system.Then we investigate the Poincarémap of each sub-smooth-system,which leads to the return map of the whole system.We prove that the interaction of the eigen-structure of each subsystem and the discontinuity determines the generalized Hopf bifurcation phenomena.
引文
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[8]V.Carmona,E.Freire,E.Ponce,J.Ros,and F.Torres.Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones.Application to Chua's circuit.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,15(10):3153-3164,2005.
[9]V.Carmona,E.Freire,E.Ponce,and F.Torres.On simplifying and classifying piecewiselinear systems.IEEE Trans.Circuits Systems I Fund.Theory Appl.,49(5):609-620,2002.
[10]V.Carmona,E.Freire,E.Ponce,and F.Torres.Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems.IMA J.Appl.Math.,69(1):71-91.2004.
[11]S.N.Chow and J.K.Hale.Methods of bifurcation theory,volume 251 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Science].Springer-Verlag,New York,1982.
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[13]M.di Bernardo,C.Budd,and A.Champneys.Grazing,skipping and sliding:analysis of the non-smooth dynamics of the DC/DC buck converter.Nonlinearity,11(4):859-890,1998.
[14]M.di Bernardo,C.Budd,and A.Champneys.Grazing,skipping and sliding:analysis of the non-smooth dynamics of the DC/DC buck converter.Nonlinearity,11(4):859-890,1998.
[15]M.di Bernardo,C.J.Budd,and A.R.Champneys.Corner collision implies bordercollision bifurcation.Phys.D,154(3-4):171-194,2001.
[16]M.di Bernardo,C.J.Budd,A.R.Champneys,and P.Kowalczyk.Piecewise-smooth dynamical systems,volume 163 of Applied Mathematical Sciences.Springer-Verlag London Ltd.,London,2008.Theory and applications.
[17]M.Di Bernardo,M.I.Feigin,S.J.Hogan,and M.E.Homer.Local analysis of Cbifurcations in n-dimensional piecewise-smooth dynamical systems.Chaos Solitons Fractals,10(11):1881-1908,1999.
[18]M.Di Bernardo,F.Garofalo,L.Iannelli,and F.Vasca.Bifurcations in piecewise-smooth feedback systems.Internat.J.Control,75(16-17):1243-1259,2002.Switched,piecewise and polytopic linear systems.
[19]M.di Bernardo,P.Kowalczyk,and A.Nordmark.Sliding bifurcations:a novel mechanism for the sudden onset of chaos in dry friction oscillators.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,13(10):2935-2948,2003.
[20]M.di Bernardo,P.Kowalczyk,and A.Nordmark.Sliding bifurcations:a novel mechanism for the sudden onset of chaos in dry friction oscillators.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,13(10):2935-2948,2003.
[21]P.Diamond,N.Kuznetsov,and D.Rachinskii.On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity.J.Differential Equations,175(1):1-26,2001.
[22]P.Diamond,D.Rachinskii,and M.Yumagulov.Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity.Nonlinear Anal.,42(6,Set.A:Theory Methods):1017-1031,2000.
[23]A.F.Filippov.Differential equations with discontinuous righthand sides,volume 18 of Mathematics and its Applications(Soviet Series).Kluwer Academic Publishers Group,Dordrecht,1988.Translated from the Russian.
[24]E.Freire,E.Ponce,F.Rodrigo,and F.Tortes.Bifurcation sets of continuous piecewise linear systems with two zones.Internal.J.Bifur.Chaos Appl.Sci.Engrg.,8(11):2073-2097,1998.
[25]E.Freire,E.Ponce,F.Rodrigo,and F.Torres.Bifurcation sets of symmetrical continuous piecewise linear systems with three zones.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,12(8):1675-1702,2002.
[26]E.Freire,E.Ponce,and J.Ros.Limit cycle bifurcation from center in symmetric piecewiselinear systems.Internal.J.Bifur.Chaos Appl.Sci.Engrg.,9(5):895-907,1999.
[27]E.Freire,E.Ponce,.and J.Ros.The focns-center-limit cycle bifurcation in symmetric 3D piecewise linear systems.SIAM J.Appl.Math.,65(6):1933-1951(electronic),2005.
[28]E.Freire,E.Ponce,and J.Ros.A biparametric bifurcation in 3D continuous piecewise linear systems with two zones.Application to Chua's circuit.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,17(2):445-457,2007.
[29]E.Freire,E.Ponce,and F.Torres.Hopf-like bifurcations in planar piecewise linear systems.In Proceedings of the Symposium on Planar Vector Fields(Lleida,1996),volume 41.pages 135-148,1997.
[30]U.Galvanetto,S.R.Bishop,and L.Briseghella.Mechanical stick-slip vibrations.Internat.J.Bifur.Chaos Appl.Sci.Engrg.,5(3):637-651,1995.
[31]F.Giannakopoulos and K.Pliete.Planar systems of piecewise linear differential equations with a line of discontinuity.Nonlinearity,14(6):1611-1632,2001.
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