两类Lienard系统的Hopf分支和一类近Hamilton系统尖点环的扰动分支
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摘要
Hilbert第16个问题的第二部分是找出任一n阶多项式系统中极限环的最大个数及其分布.很多年来,对这个问题的研究已经取得了很多的成果,特别是对二次和三次多项式系统.但是,直到现在这个问题还没有完全的解决,即使是对于n=2的情况也没有.极限环是由分支产生,这些分支包括Hopf分支、同宿分支、异宿分支、Poincare分支等.近些年来,非光滑动力系统极限环的研究也有了长足的发展,也取得了一些基础性的成果.
本论文共分五章,各章内容简介如下：
第一章为引言,主要介绍了所研究课题的来源与现状,以及本论文中所使用的研究方法和得到的主要结论.
第二章为基本引理,主要是为本论文中的几个基本引理给出了详细证明,这些引理在主要结论的证明当中起着重要作用.
第三章为光滑Lienard多项式系统的Hopf分支.首先用新的方法重新证明了Lienard多项式方程在中心点的Hopf环性数是这里qn(x)是n次多项式且qn(0)=0；然后将新方法应用于更一般的Lienard多项式系统这里pm(x)为m次多项式且pm(0)≠0,得到其在中心附近的局部极限环最大个数的上界为特别,当m=n=1、2、3、4时,我们得到系统(2)在中心点的Hopf环性数是2n-2.
第四章为非光滑Lienard系统的Hopf分支.这里我们结合使用了第二、三章中的证明方法,主要研究了非光滑多项式系统
其中得出系统(3)在原点的Hopf环性数是若m≥n,或者若n>m.
第五章为一类近Hamilton系统尖点环的扰动分支.我们主要研究近Hamilton系统这里P3(x,y)和Q3(x,y)是三次多项式,通过已知定理计算一阶Melnikov函数,从而得到该系统在尖点环附近能产生5个极限环.
本论文的主要创新在于,在系统(1)的研究中,通过变量变换证明函数的线性无关性,从而得到证明了结论所需要的条件,这与文献Petrov[11]中所用的复分析的方法有很大的不同.并且,我还将这方法成功地应用到多项式系统(2)和非光滑Lienard系统(3).
The second part of Hilbert's 16th problem is related with the maximal number and relative positions of limit cycles of polynomial systems of degree n. Many many works have been done on the study of the above problem for many years, especially for quadratic and cubic systems. However, up to now the problem has been not solved com-pletely yet even for the case of n= 2. Limit cycles are generated through bifurcation, such as Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation, Poincare bi-furcation and so on. In recent years, the study of limit cycles of nonsmooth dynamical systems has also being developed. Some general fundamental results of bifurcation for nonsmooth system were got.
This paper consists of five chapters. The particular contents of each chapter are as follows.
As an introduction, in the first chapter the background of our research and main topics, which we will study in the following chapters, are introduced. A description of our methods and results derived in this thesis can be found in this chapter.
Chapter 2 is related with preliminary lemmas. Our main purpose is to provide a detailed proof of several main lemmas which play an important role in proving main results.
In Chapter 3, we study the Hopf bifurcation of two types of smooth Lienard poly-nomial systems. First, we use a new method to prove that for the Lienard polynomial system
where qn(x) is a polynomial in the variable x of degree n and qn(0)= 0, the Hopf cyclicity near a center is Second, by applying the new method to another Lienard polynomial system
where pm(x) is a polynomial in the variable x of degree m and pm(0)≠0, we prove that for Eq.(2) the upper bound of the maximal number of local limit cycles is Further, we obtain that the Hopf cyclicity of Eq.(2) near the center is 2n - 2 for m=n=1,2,3,4.
In Chapter 4, we study the Hopf cyclicity of nonsmooth polynomial system. By applying the methods in the second and third chapter, we consider the nonsmooth polynomial system where and prove that the Hopf cyclicity at the origin is for m≥n or for n>m.
In the last chapter, we investigate one kind of near-Hamiltonian system which has a cuspidal loop and where P3(x, y) and Q3(x,y) are polynomials in the variable x and y of degree 3. And we prove that the number of limit cycles appearing in a neighborhood of the cuspidal loop is 5, by using some known bifurcation theorems to study the first Melnikov function.
The main method we use in this paper is that, during the study of Eq.(1), by mak-ing variable transformation we prove the linear independence to get the condition which is necessary to complete the proof of main results. It is different from the method of complex analysis used in Petrov [11]. And the method also can be implied to polynomial system (2) and nonsmooth Lienard system (3).
本论文共分五章,各章内容简介如下：
第一章为引言,主要介绍了所研究课题的来源与现状,以及本论文中所使用的研究方法和得到的主要结论.
第二章为基本引理,主要是为本论文中的几个基本引理给出了详细证明,这些引理在主要结论的证明当中起着重要作用.
第三章为光滑Lienard多项式系统的Hopf分支.首先用新的方法重新证明了Lienard多项式方程在中心点的Hopf环性数是这里qn(x)是n次多项式且qn(0)=0；然后将新方法应用于更一般的Lienard多项式系统这里pm(x)为m次多项式且pm(0)≠0,得到其在中心附近的局部极限环最大个数的上界为特别,当m=n=1、2、3、4时,我们得到系统(2)在中心点的Hopf环性数是2n-2.
第四章为非光滑Lienard系统的Hopf分支.这里我们结合使用了第二、三章中的证明方法,主要研究了非光滑多项式系统
其中得出系统(3)在原点的Hopf环性数是若m≥n,或者若n>m.
第五章为一类近Hamilton系统尖点环的扰动分支.我们主要研究近Hamilton系统这里P3(x,y)和Q3(x,y)是三次多项式,通过已知定理计算一阶Melnikov函数,从而得到该系统在尖点环附近能产生5个极限环.
本论文的主要创新在于,在系统(1)的研究中,通过变量变换证明函数的线性无关性,从而得到证明了结论所需要的条件,这与文献Petrov[11]中所用的复分析的方法有很大的不同.并且,我还将这方法成功地应用到多项式系统(2)和非光滑Lienard系统(3).
The second part of Hilbert's 16th problem is related with the maximal number and relative positions of limit cycles of polynomial systems of degree n. Many many works have been done on the study of the above problem for many years, especially for quadratic and cubic systems. However, up to now the problem has been not solved com-pletely yet even for the case of n= 2. Limit cycles are generated through bifurcation, such as Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation, Poincare bi-furcation and so on. In recent years, the study of limit cycles of nonsmooth dynamical systems has also being developed. Some general fundamental results of bifurcation for nonsmooth system were got.
This paper consists of five chapters. The particular contents of each chapter are as follows.
As an introduction, in the first chapter the background of our research and main topics, which we will study in the following chapters, are introduced. A description of our methods and results derived in this thesis can be found in this chapter.
Chapter 2 is related with preliminary lemmas. Our main purpose is to provide a detailed proof of several main lemmas which play an important role in proving main results.
In Chapter 3, we study the Hopf bifurcation of two types of smooth Lienard poly-nomial systems. First, we use a new method to prove that for the Lienard polynomial system
where qn(x) is a polynomial in the variable x of degree n and qn(0)= 0, the Hopf cyclicity near a center is Second, by applying the new method to another Lienard polynomial system
where pm(x) is a polynomial in the variable x of degree m and pm(0)≠0, we prove that for Eq.(2) the upper bound of the maximal number of local limit cycles is Further, we obtain that the Hopf cyclicity of Eq.(2) near the center is 2n - 2 for m=n=1,2,3,4.
In Chapter 4, we study the Hopf cyclicity of nonsmooth polynomial system. By applying the methods in the second and third chapter, we consider the nonsmooth polynomial system where and prove that the Hopf cyclicity at the origin is for m≥n or for n>m.
In the last chapter, we investigate one kind of near-Hamiltonian system which has a cuspidal loop and where P3(x, y) and Q3(x,y) are polynomials in the variable x and y of degree 3. And we prove that the number of limit cycles appearing in a neighborhood of the cuspidal loop is 5, by using some known bifurcation theorems to study the first Melnikov function.
The main method we use in this paper is that, during the study of Eq.(1), by mak-ing variable transformation we prove the linear independence to get the condition which is necessary to complete the proof of main results. It is different from the method of complex analysis used in Petrov [11]. And the method also can be implied to polynomial system (2) and nonsmooth Lienard system (3).
引文
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[4]A.Gasull, J. Torregrosa, Small-amplitude limit cycles in Lienard systems via multiplicity J. Diff. Eqns.159 (1999) 186-211.
[5]M.Han, Liapunov constants and Hopf cyclicity of Lienard systems, Ann. Diff. Eqns.15 (1999) 113-126.
[6]Jiao Jiang, Maoan Han, Small-amplitude limit cycles of some Lienard-type systems Non-linear Analysis,71 (2009) 6373-6377
[7]J. Jiang, M. Han, P. Yu, S. Lynch, Limit cycles in two types of symmetric Lienard's systems, Inter. J. Bifurcation and Chaos 17 (2007) 2169-2174.
[8]A. Lins, W. deMelo, C. Pugh, On equation, Geometry and topology, Lecture notes in Math. Springer Berlin Vol.597 (1977) 335-357.
[9]N. Lloyd, S. Lynch, Small-amplitude limit cycles of certain Lienard systems, Proc. Royal Society of London (A) 418 (1988) 199-208.
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[14]P.Yu, M. Han, On limit cycles of the Li enard equations with Z2 symmetry, Chaos Solit. Fract.30 (2006) 1048-1068.
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[33]M. Han, T. Zhang, Some bifurcation methods of finding limit cycles, Math. Biosci. Eng. 3(1)(2006) 67-77.
[34]M. Han, H. Zang, J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations 246 (2009) 129-163
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[2]C.J. Christopher, N. G. Lloyd, Smallamplitude limit cycles in polynomial Lienard sys-tems, Nonlin. Diff. Eqns. Appl.3 (1996) 183-190.
[3]C. Christopher, S. Lynch, Small-amplitude limit cycle bifurcations for Lienard systems with quadratic or cubic damping or restoring forces, Nonlinearity 12 (1999) 1099-1112.
[4]A.Gasull, J. Torregrosa, Small-amplitude limit cycles in Lienard systems via multiplicity J. Diff. Eqns.159 (1999) 186-211.
[5]M.Han, Liapunov constants and Hopf cyclicity of Lienard systems, Ann. Diff. Eqns.15 (1999) 113-126.
[6]Jiao Jiang, Maoan Han, Small-amplitude limit cycles of some Lienard-type systems Non-linear Analysis,71 (2009) 6373-6377
[7]J. Jiang, M. Han, P. Yu, S. Lynch, Limit cycles in two types of symmetric Lienard's systems, Inter. J. Bifurcation and Chaos 17 (2007) 2169-2174.
[8]A. Lins, W. deMelo, C. Pugh, On equation, Geometry and topology, Lecture notes in Math. Springer Berlin Vol.597 (1977) 335-357.
[9]N. Lloyd, S. Lynch, Small-amplitude limit cycles of certain Lienard systems, Proc. Royal Society of London (A) 418 (1988) 199-208.
[10]D. Luo, X. Wang, D. Zhu, M. Han, Bifurcation Theory and Methods of Dynamical Sys-tems, Advanced Series in Dynamical Systems Vol.15 (1997) World Scientific Singapore.
[11]G. Petrov, The Chebyshev property of elliptic integrals, Funct. Anal.,22(1988),72-73.
[12]D. Xiao, Bifurcations on a five-parameter family of planar vector field. J. Dynam. Differ-ential Equations 20 (2008), no.4,961-980.
[13]D. Xiao, H. Zhu, Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math.66 (2006), no.3,802-819.
[14]P.Yu, M. Han, On limit cycles of the Li enard equations with Z2 symmetry, Chaos Solit. Fract.30 (2006) 1048-1068.
[15]X. Liu, M. Han, Hopf bifurcation for nonsmooth lienard system, International Journal of Bifurcation and Chaos, Vol.19, no.7 (2009) 2401-2415.
[16]Coll, B., Gasull, A.& Prohens, R. [1996] "Limit cycles for non smooth differential equa-tions via schwarzian derivative," J. Diff. Eqs.132,203-221.
[17]Coll, B., Prohens, R.& Gasull, A. [1999] "The center problem for discontinuous Lienard differential equation," Int. J. Bifurcation and Chaos 9,1751-1761.
[18]Coll, B., Gasull, A.& Prohens, R. [2000] "Center-focus and isochronous center problems for discontinuous differential equations," Discr. Contin. Dyn. Syst.6,609-624.
[19]Coll, B., Gasull, A.& Prohens, R. [2001] "Degenerate Hopf bifurcation in discontinuous planar systems," J. Math, Anal.Appl.253,671-690.
[20]Filippov, A. F. [1988] Differential Equations with Discontinuous Bighthand Sides (Kulwer Academic Pub.,Netherlands).
[21]Gasull, A.& Torregrosa, J. [2003] "Center-focus problem for discontinuous planar differ ential equations," Int. J. Bifurcation and Chaos 13,1755-1765.
[22]Han, M. [2000] "On Hopf cyclicity of planar systems," J. Math. Anal. Appl.245,404-422.
[23]Kunze, M. [2000] Non-smooth dynamical systems (Springer-Verlag, Berlin).
[24]Kiipper, T.& Moritz, S. [2001] "Generalized Hopf bifurcation for non-smooth planar systems," Phil. Trans. R. Soc. Lond. A 359,2483-2496.
[25]Leine, R. I. et al. [2000] "Bifurcations in nonlinear discontinuous systems," Nonlin. Dyn. 23,105-164.
[26]Leine, R. I.,& Van Campen, D. H. [2006] "Bifurcation phenomena in non-smooth dy-namical systems," European Journal of Mechanics A/Solids 25,595-616.
[27]Leine, R. I. [2006] "Bifurcations of equilibria in non-smooth continuous systems," Physica D 223,121-137.
[28]Zou, Y., Kiipper, T.& Beyn, W. J. [2006] "Generalized Hopf bifurcation for palnar Filippov systems continuous at the origin," J. Nonlinear Sci.16,159-177.
[29]F. Dumortier, C. Li, Perturbations from an elliptic Hamiltonian of degree four, Ⅱ. Cus-pidal loop. J. Differential Equations 175(2)(2001) 209-243.
[30]M. Han, Bifurcation Theory of Limit Cycles of Planar Systems, Handbook of Differential Equations, Ordinary Differential Equations, vol.3, Edited by A. Canada, P. Drabek and A. Fonda,2006, Elsevier.
[31]M. Han, J. Chen, On the number of limit cycles in double homoclinic bifurcations, Sci. China Ser. A 43(9) (2000) 914-928.
[32]M. Han, J. Yang, A. Tarta, Y. Gao, Limit cycles near homoclinic and heteroclinic loops, to appear in JDDE.
[33]M. Han, T. Zhang, Some bifurcation methods of finding limit cycles, Math. Biosci. Eng. 3(1)(2006) 67-77.
[34]M. Han, H. Zang, J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations 246 (2009) 129-163
[35]R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Brasil. Mat.17(2) (1986) 67-101.
[36]D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and Periodic Orbits of Vector Fields, ed. Schlomiuk, D., NATO ASI Series C, Vol.408(1993) (Kluwer Academic, London),429-467.
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