二阶次线性方程的无穷多次调和解
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摘要
本文考虑被数学和其他科学、工程领域的研究人员广为关注的两个重要的二阶次线性微分方程模型：具有有界恢复力的Duffing方程和次线性碰撞振子的无穷多个次调和解的存在性。
对于有界恢复力的Duffing方程，我们首先在原点附近对原方程进行改造，使之成为一个新的平面Hamilton系统，且保证(0，0)是该Hamilton系统的解。再通过一个函数来控制内圈，保证其Pioncaré映射满足边界扭转条件。然后用丁伟岳推广的PoincaréBirkhoff扭转定理得到新系统次调和解的存在性和多重性，而不动点对应的扭转角度又保证了这些次调和解恰好也是原方程的次调和解，最后用Massera定理得到原方程调和解的存在性。
对于次线性碰撞振子，我们首先引进新的坐标变换把右半平面上的碰撞问题转化到整个平面上，且将碰撞系统转化为与之等价的新系统．然后采用与讨论次线性Duffing方程类似的思想来处理新的碰撞系统，从而得到原碰撞振子的无穷多次调和弹性解的存在性。
In this article, We consider two important models for second order sublinear differential equations: the existence of infinitely many subharmonic solutions for Duffing equation with bounded restore force and for the sublinear impact oscillator. They have attracted lots of researchers' attention in mathematical, other scientific and engineering fields.
As for Duffing equation with bounded restore force, firstly we will change the old equation in a neighborhood of (0, 0). We change it into a new plane Hamiltonian system and make sure that (0, 0) is a solution for the Hamiltonian system. Then we use a function to control the inner boundary and make the Poincarémapping have the property of boundary twist. We obtain the existence of infinitely many subharmonic solutions for the new system by using Ding's Poincaré-Birkhoff theorem; and because of the angles of rotation for these subharmonic solutions, they are just the subharmonic solutions for the old equation. At last, we get a harmonic solution for the old equation by Massera theorem.
As for the sublinear impact oscillator, firstly we will introduce a new coordinate transformation. Ittransforms the old impact system from right half plane to the whole plane, and translates the system into a new equal system. Then we deal with the new system with a method similar to the one we have applied to the sublinear equation. At last, we can get infinitely many subharmonic bouncing solutions for the old impact system.
对于有界恢复力的Duffing方程，我们首先在原点附近对原方程进行改造，使之成为一个新的平面Hamilton系统，且保证(0，0)是该Hamilton系统的解。再通过一个函数来控制内圈，保证其Pioncaré映射满足边界扭转条件。然后用丁伟岳推广的PoincaréBirkhoff扭转定理得到新系统次调和解的存在性和多重性，而不动点对应的扭转角度又保证了这些次调和解恰好也是原方程的次调和解，最后用Massera定理得到原方程调和解的存在性。
对于次线性碰撞振子，我们首先引进新的坐标变换把右半平面上的碰撞问题转化到整个平面上，且将碰撞系统转化为与之等价的新系统．然后采用与讨论次线性Duffing方程类似的思想来处理新的碰撞系统，从而得到原碰撞振子的无穷多次调和弹性解的存在性。
In this article, We consider two important models for second order sublinear differential equations: the existence of infinitely many subharmonic solutions for Duffing equation with bounded restore force and for the sublinear impact oscillator. They have attracted lots of researchers' attention in mathematical, other scientific and engineering fields.
As for Duffing equation with bounded restore force, firstly we will change the old equation in a neighborhood of (0, 0). We change it into a new plane Hamiltonian system and make sure that (0, 0) is a solution for the Hamiltonian system. Then we use a function to control the inner boundary and make the Poincarémapping have the property of boundary twist. We obtain the existence of infinitely many subharmonic solutions for the new system by using Ding's Poincaré-Birkhoff theorem; and because of the angles of rotation for these subharmonic solutions, they are just the subharmonic solutions for the old equation. At last, we get a harmonic solution for the old equation by Massera theorem.
As for the sublinear impact oscillator, firstly we will introduce a new coordinate transformation. Ittransforms the old impact system from right half plane to the whole plane, and translates the system into a new equal system. Then we deal with the new system with a method similar to the one we have applied to the sublinear equation. At last, we can get infinitely many subharmonic bouncing solutions for the old impact system.
引文
[1] T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations, 105(1993), 364-409.
[2] D. Qian, Time maps and Duffing equations acrossing resonance, Science in China, 23A(1993), 471-479.
[3] D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer. Leach-Dancer condition, J. Differential Equation, 171(2001), 233-250.
[4] 丁同仁，常微分方程定性方法的应用，高等教育出版社，北京，2004．
[5] 丁同仁，关于亚线性Duffing方程的研究，应用数学学报，12(1989)，449-455．
[6] T. Ding and F. Zanolin, Subharmonic solution of second order nonlinear equations: a time-map approach, Nonlinear Analysis TMA, 20(1993), 509-532.
[7] 丁伟岳，扭转映射的不动点和常微分方程的周期解，数学学报，25(1982)，227-235．
[8] 魏兰阁，一个Duffing方程的调和解和次调和解，数学进展，32(2003)，39-46．
[9] X. Li, Boundedness of solutions for a class of Duffing equation with semilinear potential, J. Differential Equation, 176(2001), 248-268.
[10] Y. Wang, Boundedness of solutions in a class of Duffing equation with a bounded restore force, Dynamical systems, 14(2006), 783-800.
[11] H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Phys. D., 82(1995), 117-135.
[12] P. Boyland, Dual billards, twist maps and impact oscillators, Nonlinearity, 9(1996), 1411-1438.
[13] M. Corbera and J. Liibre, Periodic orbits of a conlinear restricted three body problem, Celestial Mech. Dyam. Astronom, to appear.
[14] M. Kunze, Non-smooth dynamical systems, Lecture Notes in Math. 1744, Spring-Verlag, New York, 2000.
[15] M. Kunze and T. Küpper, Non-smooth dynamical systems: an overview, In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (edited by B. Fieldler), Springer-Verlag, Berlin-Heidelberg-New York, 2001, 431-452.
[16] Albert, C. J. Luo and L. Chen, Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons and Fractals, 24(2005), 567-578.
[17] C. Bapat, Periodic motions of an impact oscillator, J..Sound and Vibration, 209(1998), 43-60.
[18] B. Clark, E. Rosa, A. Hall and T. Shepherd, Dynamics of an electronic impact oscillator, Physics Letters A, 318(2003), 514-521.
[19] E. Pavlovskaia and M. Wiercigroch, Periodic solution finder for an impact oscillator with a drift, J. Sound and Vibration, 267(2003), 893-911.
[20] D. Bonheune and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bouncing, Nonlinearity, 15(2002), 1281-1298.
[21] A. Lazer and P. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differential and Integral Equations, 5(1992), 165-172.
[22] D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, proceeding of the Royal Society of Edinburgh, 134A(2004), 201-213.
[23] D. Qian and P. J. Tortes, Periodic motions of linear impact oscillators via successor map, SIAM J. Math. Anal., 36(2005), 1707-1725.
[24] W. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88(1983), 341-346.
[25] J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128(1998), 139-151.
[26] X. Sun and D. Qian, Impact oscillators and characteristic value, to appear.
[27] J. Hale, Ordinary differential equation, Robert E. Krieger Publishing Campany Huntingto, New York, i980.
[28] H. Jacobowitz, Periodic solutions o.f x" + f(x,t) = 0 via the Poincaré-Birkhoff theorem, J. Differential Equations, 20(1976), 37-52.
[29] R. Ortega, Asymmetric oscillators and twist mapping, J. Lond. Math. Soc., 53(1996), 325-342.
[30] 孙西滢，弹性碰撞振子的动态行为，苏州大学硕士学位论文，2003．
[2] D. Qian, Time maps and Duffing equations acrossing resonance, Science in China, 23A(1993), 471-479.
[3] D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer. Leach-Dancer condition, J. Differential Equation, 171(2001), 233-250.
[4] 丁同仁，常微分方程定性方法的应用，高等教育出版社，北京，2004．
[5] 丁同仁，关于亚线性Duffing方程的研究，应用数学学报，12(1989)，449-455．
[6] T. Ding and F. Zanolin, Subharmonic solution of second order nonlinear equations: a time-map approach, Nonlinear Analysis TMA, 20(1993), 509-532.
[7] 丁伟岳，扭转映射的不动点和常微分方程的周期解，数学学报，25(1982)，227-235．
[8] 魏兰阁，一个Duffing方程的调和解和次调和解，数学进展，32(2003)，39-46．
[9] X. Li, Boundedness of solutions for a class of Duffing equation with semilinear potential, J. Differential Equation, 176(2001), 248-268.
[10] Y. Wang, Boundedness of solutions in a class of Duffing equation with a bounded restore force, Dynamical systems, 14(2006), 783-800.
[11] H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Phys. D., 82(1995), 117-135.
[12] P. Boyland, Dual billards, twist maps and impact oscillators, Nonlinearity, 9(1996), 1411-1438.
[13] M. Corbera and J. Liibre, Periodic orbits of a conlinear restricted three body problem, Celestial Mech. Dyam. Astronom, to appear.
[14] M. Kunze, Non-smooth dynamical systems, Lecture Notes in Math. 1744, Spring-Verlag, New York, 2000.
[15] M. Kunze and T. Küpper, Non-smooth dynamical systems: an overview, In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (edited by B. Fieldler), Springer-Verlag, Berlin-Heidelberg-New York, 2001, 431-452.
[16] Albert, C. J. Luo and L. Chen, Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons and Fractals, 24(2005), 567-578.
[17] C. Bapat, Periodic motions of an impact oscillator, J..Sound and Vibration, 209(1998), 43-60.
[18] B. Clark, E. Rosa, A. Hall and T. Shepherd, Dynamics of an electronic impact oscillator, Physics Letters A, 318(2003), 514-521.
[19] E. Pavlovskaia and M. Wiercigroch, Periodic solution finder for an impact oscillator with a drift, J. Sound and Vibration, 267(2003), 893-911.
[20] D. Bonheune and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bouncing, Nonlinearity, 15(2002), 1281-1298.
[21] A. Lazer and P. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differential and Integral Equations, 5(1992), 165-172.
[22] D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, proceeding of the Royal Society of Edinburgh, 134A(2004), 201-213.
[23] D. Qian and P. J. Tortes, Periodic motions of linear impact oscillators via successor map, SIAM J. Math. Anal., 36(2005), 1707-1725.
[24] W. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88(1983), 341-346.
[25] J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128(1998), 139-151.
[26] X. Sun and D. Qian, Impact oscillators and characteristic value, to appear.
[27] J. Hale, Ordinary differential equation, Robert E. Krieger Publishing Campany Huntingto, New York, i980.
[28] H. Jacobowitz, Periodic solutions o.f x" + f(x,t) = 0 via the Poincaré-Birkhoff theorem, J. Differential Equations, 20(1976), 37-52.
[29] R. Ortega, Asymmetric oscillators and twist mapping, J. Lond. Math. Soc., 53(1996), 325-342.
[30] 孙西滢，弹性碰撞振子的动态行为，苏州大学硕士学位论文，2003．