Duffing型方程的Aubry-Mather集
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摘要
Duffing型方程是非线性Hamilton系统重要的模型之一,由于其在实际模型中具有代表性和数学研究的可行性,其动力学行为一直受到人们普遍关注。这篇文章我们考虑Duffimg型方程的Aubry-Mather集存在性问题。
在文章第一部分里,我们首先引进类似于极坐标的坐标变换,对变换后的方程进行一系列细致估计,证明其Poincaré映射具有单调扭转性质;再运用裴明亮推广的Aubry-Mather集定理,得到了一类半线性Duffing方程Aubry-Mather集的存在性。
在文章第二部分里,我们首先推广了第一部分的坐标变换,使得该新的变换适合次线性Duffing方程模型,对变换后的方程进行细致估计,证明其Poincaré映射具有单调扭转性质,从而运用裴明亮推广的Aubry-Mather集定理证得一类次线性Duffing型方程存在Aubry-Mather集。
在文章第三部分里,我们首先引进新的坐标变换把半线性Duffing方程右半平面上的碰撞问题转化到整个平面上,对变换后的方程的Poincaré映射进行细致估计,证明其具有单调扭转性质,最后化归为可以应用钱定边推广的Aubry-Mather集定理的情况,得到了一类半线性Duffing方程碰撞振子的Aubry-Mather集的存在性。
Duffing-type equation is one of important models in nonlinear Hamilton systen, its dynamical behaviour has been widely investigated in the literature because of its significance for the applications as well as for its mathematical fascination. In this thesis, we consider the existence of the Aubry-Mather sets for Duffing-type equation.In section 1, we first introduce a new coordinate transformation which is similar to polor coordinate transformation. With careful estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property , then based on the Aubry-Mather theorem generalized by Pei, we get the existence of the Aubry-Mather sets for a class of semilinear Duffing equations.In section 2, we extend the coordinate transformation in section 1 which is suitable to sublinear Duffing-type equation. With careful estimates after action-angle variable transformation, we prove that the Poincare map is a monotone map. then applying the Aubry-Mather theorem generalized by Pei, we obtain the existence of the Aubry-Mather sets for a class of sublinear Duffing-type equations.In section 3, at the beginning, we introduce a new coordinate transformation which changes the semilinear Duffing-type equations with bouncing from right half plane into the whole plane except the origin. With some delicate estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property. Then we obtain the existence of the Aubry-Mather sets of semilinear Duffing equation with bouncing via the the Aubry-Mather theorem generalized generalized by Qian.
在文章第一部分里,我们首先引进类似于极坐标的坐标变换,对变换后的方程进行一系列细致估计,证明其Poincaré映射具有单调扭转性质;再运用裴明亮推广的Aubry-Mather集定理,得到了一类半线性Duffing方程Aubry-Mather集的存在性。
在文章第二部分里,我们首先推广了第一部分的坐标变换,使得该新的变换适合次线性Duffing方程模型,对变换后的方程进行细致估计,证明其Poincaré映射具有单调扭转性质,从而运用裴明亮推广的Aubry-Mather集定理证得一类次线性Duffing型方程存在Aubry-Mather集。
在文章第三部分里,我们首先引进新的坐标变换把半线性Duffing方程右半平面上的碰撞问题转化到整个平面上,对变换后的方程的Poincaré映射进行细致估计,证明其具有单调扭转性质,最后化归为可以应用钱定边推广的Aubry-Mather集定理的情况,得到了一类半线性Duffing方程碰撞振子的Aubry-Mather集的存在性。
Duffing-type equation is one of important models in nonlinear Hamilton systen, its dynamical behaviour has been widely investigated in the literature because of its significance for the applications as well as for its mathematical fascination. In this thesis, we consider the existence of the Aubry-Mather sets for Duffing-type equation.In section 1, we first introduce a new coordinate transformation which is similar to polor coordinate transformation. With careful estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property , then based on the Aubry-Mather theorem generalized by Pei, we get the existence of the Aubry-Mather sets for a class of semilinear Duffing equations.In section 2, we extend the coordinate transformation in section 1 which is suitable to sublinear Duffing-type equation. With careful estimates after action-angle variable transformation, we prove that the Poincare map is a monotone map. then applying the Aubry-Mather theorem generalized by Pei, we obtain the existence of the Aubry-Mather sets for a class of sublinear Duffing-type equations.In section 3, at the beginning, we introduce a new coordinate transformation which changes the semilinear Duffing-type equations with bouncing from right half plane into the whole plane except the origin. With some delicate estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property. Then we obtain the existence of the Aubry-Mather sets of semilinear Duffing equation with bouncing via the the Aubry-Mather theorem generalized generalized by Qian.
引文
[1] Aubry S. and Le Daeron P., The discrete Frenkel Kontorova model and its extensions: Exact results for the ground state, Physica D (1983) 8, 240-258.
[2] Mather J., existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, (1982)21, 457-467.
[3] Katok A., Some remarks on Birkhoff and Mather twist maps theorem, Ergodic Theory Dynamical Systems (1982) 2, 183-194.
[4] 蒋美跃,裴明亮,扭转映射的Aubry-Mather理论及其应用,数学进展(1994)Vol.23 No.2.97-114.
[5] Denzler J., Mather sets for plane Hamiltonian systems, ZAMP (1987) 38,791-812.
[6] 裴明亮,超线性Duffing方程的Mather集,中国科学(A辑)(1993)Vol.23 No.1,21-30.
[7] Qian D., Mather sets for sublinear Duffing Equations, Chin. Ann. of Math. (1994) 15B: 4, 421-434.
[8] Pei L., Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations (1994) 113, 106-127.
[9] Arnold V. I., "Mathematical Methods of Classical Mechanics", Springer-Verlag, New York, 1978.
[10] Capietto A. and Liu B., Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis (2004) 56, 105-117.
[11] Levi M., Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. (1991) 143, 43-83.
[12] Kunze M., Küpper T. and You J., On the application of KAM theorey to discontinuous dynamical systems, J. Differential Equations (1997) 139, 1-21.
[13] Liu B., Boundedness in nonlinear oscillations at resonance, J. Differential Equations (1999) 153, 142-154.
[14] Zharnitsky V., Invariant tori in Hamiltionian systems with impacts, Comm. Math. Phys. (2000) 211, 289-302.
[15] Kunze M., Non-Smooth Dynamical Systems, Lecture Notes in Math. 1744, Spring-Verlag, New York, 2000.
[16] Kunze M. and Kǜpper T., 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.
[17] Bapat C., Periodic motions of an impact oscillator, J. Sound and Vibration (1998)209, 43-60.
[18] Clark B., Rosa E., Hall A., Shepherd T., Dynamics of an electronic impact oscillator, Physics Letters A (2003)318, 514-521.
[19] Pavlovskaia E. and Wiercigroch M., Periodic solution finder for an impact oscillator with a drift, J. Sound and Vibration(2003)267, 893-911.
[20] Albert, Luo C. J. and Chen L., Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons and Fractals (2005) 24, 567-578.
[21] 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.
[22] Hale J., Ordinary differential equation, Robert E. Krieger Publishing Campany Huntingto, New York, 1980.
[2] Mather J., existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, (1982)21, 457-467.
[3] Katok A., Some remarks on Birkhoff and Mather twist maps theorem, Ergodic Theory Dynamical Systems (1982) 2, 183-194.
[4] 蒋美跃,裴明亮,扭转映射的Aubry-Mather理论及其应用,数学进展(1994)Vol.23 No.2.97-114.
[5] Denzler J., Mather sets for plane Hamiltonian systems, ZAMP (1987) 38,791-812.
[6] 裴明亮,超线性Duffing方程的Mather集,中国科学(A辑)(1993)Vol.23 No.1,21-30.
[7] Qian D., Mather sets for sublinear Duffing Equations, Chin. Ann. of Math. (1994) 15B: 4, 421-434.
[8] Pei L., Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations (1994) 113, 106-127.
[9] Arnold V. I., "Mathematical Methods of Classical Mechanics", Springer-Verlag, New York, 1978.
[10] Capietto A. and Liu B., Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis (2004) 56, 105-117.
[11] Levi M., Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. (1991) 143, 43-83.
[12] Kunze M., Küpper T. and You J., On the application of KAM theorey to discontinuous dynamical systems, J. Differential Equations (1997) 139, 1-21.
[13] Liu B., Boundedness in nonlinear oscillations at resonance, J. Differential Equations (1999) 153, 142-154.
[14] Zharnitsky V., Invariant tori in Hamiltionian systems with impacts, Comm. Math. Phys. (2000) 211, 289-302.
[15] Kunze M., Non-Smooth Dynamical Systems, Lecture Notes in Math. 1744, Spring-Verlag, New York, 2000.
[16] Kunze M. and Kǜpper T., 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.
[17] Bapat C., Periodic motions of an impact oscillator, J. Sound and Vibration (1998)209, 43-60.
[18] Clark B., Rosa E., Hall A., Shepherd T., Dynamics of an electronic impact oscillator, Physics Letters A (2003)318, 514-521.
[19] Pavlovskaia E. and Wiercigroch M., Periodic solution finder for an impact oscillator with a drift, J. Sound and Vibration(2003)267, 893-911.
[20] Albert, Luo C. J. and Chen L., Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons and Fractals (2005) 24, 567-578.
[21] 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.
[22] Hale J., Ordinary differential equation, Robert E. Krieger Publishing Campany Huntingto, New York, 1980.