格点系统和二阶变号位势方程的周期解和相关动力行为
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摘要
格点系统和带变号位势的二阶微分方程是两类重要的微分方程模型。本文讨论耦合格点系统和二阶变号位势方程的周期解和相关动力行为,主要考虑:带阻尼的次线性耦合受迫格点系统的周期解;二阶超线性变号位势方程的周期解和周期碰撞解;二阶超线性变号位势碰撞方程的全局定义解以及混沌行为。
在第一章,阐述了课题的背景以及意义,介绍了格点系统和变号位势二阶微分方程解的动力行为的研究情况与本文的主要研究成果。
在第二章,考虑一类粒子间相邻耦合的非保守受迫格点系统,其中粒子间的耦合力关于粒子间的距离是次线性增长的。当系统是有限维时,我们通过寻找同伦方程组的周期解的先验界和运用拓扑度的方法证明了周期解存在的充分必要条件;当系统是无穷维时,我们在有限维结果的基础之上运用一些极限讨论得到了无穷多个周期解的存在性。
在第三章,考虑一类非保守超线性变号位势方程的周期解。首先,我们对所谓“拉伸-扭转映射”给出一个新的拓扑不动点定理。其次,针对权函数取负值和非负值两种情况对解的动力行为分别进行分析。我们得出,当权函数取负值时解表现出快速拉伸性和部分解的不可延拓性等特点,当权函数取非负值时解具有弹性性质并且大范数解具有快速扭转性。最后,利用上述解在相平面上的定性性质,我们构造一系列适当的拓扑四边形使得在这些四边形上Poincare映射是拉伸-扭转映射,从而运用拉伸-扭转不动点定理可以证明无穷多个周期解的存在性。
在第四章,运用相平面定性分析的方法,我们考虑一类超线性受迫变号位势方程的碰撞周期解的存在性和多解性。首先,通过使用一个截断函数来构造一个新碰撞方程,使得新方程在原点附近的动力行为是简单的。这样,在原点附近,我们可以避免讨论由于强迫项的出现而导致的异常复杂的解的动态行为。同时,在适当大的圆周外面,新方程与原方程是等价的。其次,通过引进新的坐标变换我们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的系统,通过证明新的系统连续周期解的存在性来得出碰撞问题周期解的存在性。最后,运用第三章中的方法,通过对新系统的解在新的相平面上的定性分析,我们构造一系列适当的拓扑四边形并且运用拉伸-扭转不动点定理证明了无穷多个2π-周期解的存在性,从而得到原系统的无穷多个2π-周期碰撞解的存在性。
在第五章,考虑了一类超线性变号位势弹性碰撞振子的全局解和混沌动力行为,其中它的权函数在定义区间上可以无穷次变号。首先,利用第四章的坐标变换,我们将碰撞问题转化到全平面上进行讨论。其次,运用相平面定性分析方法,对权函数有限次变号的情况进行讨论,得到了在q有限次变号区间上的解的存在性,并且这些解在q不同符号的区间上有相应的碰撞次数,这个结论在后面全局定义解的讨论中是重要的。随后,通过对解在权函数取不同符号的区间上的动力行为的细致分析,同时运用了一些简单拓扑知识,我们证明了:对于任意一个事先适当选取的无穷维向量,都存在一个全局定义的碰撞解,使得在每一个权函数取非负数或取负数的区间上,解发生碰撞的次数都与这个向量对应位置上的分量相同。最后,当碰撞系统是周期系统时,利用前面所得到的全局定义解的结果,我们还证明了系统具有混沌动力行为的特征。
Lattices and second order equations with indefinite weight are two important models of dynamical systems and differential equations. In this paper, we study the periodic solutions and related dynamics of the solutions of lattices and second order equations with indefinite weight. We will study them in the following parts: the existence of periodic solutions of a class of sub-linear lattices with nearest neighbour interaction; the existence of periodic solutions and periodic impact solutions for some class of super-linear equations with indefinite weight; the existence and multiplicity of global-defined bouncing solutions and chaotic dynamics of a class of super-linear oscillator with indefinite weight.
In Chapter 1, we introduce the background and the recent reserches of these topics. Then state the main results of this paper.
In Chapter 2, we are concerned with the existence of T—periodic solutions of some non-conservative systems of classic particals periodically pertured with nearest neighbor coupling and the restoring forces are sub-linear about the distance between particles. By using a priori bounds and topological degree, over the mean values of the external forces, we find a necessary and sufficient conditions for the existence of periodic solutions in the case of finite systems and at the same time, we prove the existence of infinite periodic solutions in the case of infinite systems.
In Chapter 3, we consider the existence of periodic solutions to a class of second order non-conservative super-linear equations with indefinite weight. First of all, we present a new so called bend-twist fixed point theorem. Secondly, we investigate the dynamics of the solutions on the phase plane and we find that, the solutions with big norms have a strong oscillations in the intervals where the weight is positive as well as the blow-up phenomene are appear in the intervals where the weight is negative. Finally, based on the qualitative properties of solutions, we apply the bend-twist theorem on a series of topological quadrangles suitable constructed and we prove the existence of infinite periodic solutions for the equations.
In Chapter 4, by using the qualitative analysis in phase-plane, we consider the existence and the multiplicity of periodic bouncing solutions to a forced super-linear oscillators with indefinite weight. Firstly, by using a truncation function, we define a new impact equations such that the dynamics of the solutions is simple in a neighbourhood of the origin so that we can avoid the arguments for complicated dynamical behaviour of the solutions produced by the forced term nearby the origin. Secondly, we will introduce a new coordinate transformation, transform the impact phase-plane from right half plane to the whole plane. Thus we can use the similar arguments in In Chapter 3 to construct a seies of topological quadrangles and apply the bend-twist theorm to obtain the existence of infinite periodic solutions. These periodic solutions has large norm, therefore we prove the existence of infinite periodic bouncing solutions for the original impact oscillators.
In Chapter 5, we discuss the existence of the global defined solutions and chaotic dynamics of solutions to a class of super-linear impact oscillators with indefinite weight. At first, doing as in Chapter 4, we introduce a new coordinate transformation to translate the impact systems into a new equal systems which defined in the whole phase-plane. Second, by using the qualitative method, we get the existence of solutions that defined in the intervals in which the weight change sign for finite times. And in this case, the solutions have different impact times given beforehand in the intervals where the weight take positive number or negative number. This fact help to obtain the existence of global defined solutions. Latter, by some delicate analysis for the dynamics of solutions in the phase plane, we prove that, for each infinite dimentional non-negative integer vector there is a global defined impact solution such that the impact times of the solution in each positive weight interval or negative weight interval, are the same as the number in the corresponding component. When the weight is periodic, above results imply that the impact oscillator exhibits chaotic-like dynamics.
在第一章,阐述了课题的背景以及意义,介绍了格点系统和变号位势二阶微分方程解的动力行为的研究情况与本文的主要研究成果。
在第二章,考虑一类粒子间相邻耦合的非保守受迫格点系统,其中粒子间的耦合力关于粒子间的距离是次线性增长的。当系统是有限维时,我们通过寻找同伦方程组的周期解的先验界和运用拓扑度的方法证明了周期解存在的充分必要条件;当系统是无穷维时,我们在有限维结果的基础之上运用一些极限讨论得到了无穷多个周期解的存在性。
在第三章,考虑一类非保守超线性变号位势方程的周期解。首先,我们对所谓“拉伸-扭转映射”给出一个新的拓扑不动点定理。其次,针对权函数取负值和非负值两种情况对解的动力行为分别进行分析。我们得出,当权函数取负值时解表现出快速拉伸性和部分解的不可延拓性等特点,当权函数取非负值时解具有弹性性质并且大范数解具有快速扭转性。最后,利用上述解在相平面上的定性性质,我们构造一系列适当的拓扑四边形使得在这些四边形上Poincare映射是拉伸-扭转映射,从而运用拉伸-扭转不动点定理可以证明无穷多个周期解的存在性。
在第四章,运用相平面定性分析的方法,我们考虑一类超线性受迫变号位势方程的碰撞周期解的存在性和多解性。首先,通过使用一个截断函数来构造一个新碰撞方程,使得新方程在原点附近的动力行为是简单的。这样,在原点附近,我们可以避免讨论由于强迫项的出现而导致的异常复杂的解的动态行为。同时,在适当大的圆周外面,新方程与原方程是等价的。其次,通过引进新的坐标变换我们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的系统,通过证明新的系统连续周期解的存在性来得出碰撞问题周期解的存在性。最后,运用第三章中的方法,通过对新系统的解在新的相平面上的定性分析,我们构造一系列适当的拓扑四边形并且运用拉伸-扭转不动点定理证明了无穷多个2π-周期解的存在性,从而得到原系统的无穷多个2π-周期碰撞解的存在性。
在第五章,考虑了一类超线性变号位势弹性碰撞振子的全局解和混沌动力行为,其中它的权函数在定义区间上可以无穷次变号。首先,利用第四章的坐标变换,我们将碰撞问题转化到全平面上进行讨论。其次,运用相平面定性分析方法,对权函数有限次变号的情况进行讨论,得到了在q有限次变号区间上的解的存在性,并且这些解在q不同符号的区间上有相应的碰撞次数,这个结论在后面全局定义解的讨论中是重要的。随后,通过对解在权函数取不同符号的区间上的动力行为的细致分析,同时运用了一些简单拓扑知识,我们证明了:对于任意一个事先适当选取的无穷维向量,都存在一个全局定义的碰撞解,使得在每一个权函数取非负数或取负数的区间上,解发生碰撞的次数都与这个向量对应位置上的分量相同。最后,当碰撞系统是周期系统时,利用前面所得到的全局定义解的结果,我们还证明了系统具有混沌动力行为的特征。
Lattices and second order equations with indefinite weight are two important models of dynamical systems and differential equations. In this paper, we study the periodic solutions and related dynamics of the solutions of lattices and second order equations with indefinite weight. We will study them in the following parts: the existence of periodic solutions of a class of sub-linear lattices with nearest neighbour interaction; the existence of periodic solutions and periodic impact solutions for some class of super-linear equations with indefinite weight; the existence and multiplicity of global-defined bouncing solutions and chaotic dynamics of a class of super-linear oscillator with indefinite weight.
In Chapter 1, we introduce the background and the recent reserches of these topics. Then state the main results of this paper.
In Chapter 2, we are concerned with the existence of T—periodic solutions of some non-conservative systems of classic particals periodically pertured with nearest neighbor coupling and the restoring forces are sub-linear about the distance between particles. By using a priori bounds and topological degree, over the mean values of the external forces, we find a necessary and sufficient conditions for the existence of periodic solutions in the case of finite systems and at the same time, we prove the existence of infinite periodic solutions in the case of infinite systems.
In Chapter 3, we consider the existence of periodic solutions to a class of second order non-conservative super-linear equations with indefinite weight. First of all, we present a new so called bend-twist fixed point theorem. Secondly, we investigate the dynamics of the solutions on the phase plane and we find that, the solutions with big norms have a strong oscillations in the intervals where the weight is positive as well as the blow-up phenomene are appear in the intervals where the weight is negative. Finally, based on the qualitative properties of solutions, we apply the bend-twist theorem on a series of topological quadrangles suitable constructed and we prove the existence of infinite periodic solutions for the equations.
In Chapter 4, by using the qualitative analysis in phase-plane, we consider the existence and the multiplicity of periodic bouncing solutions to a forced super-linear oscillators with indefinite weight. Firstly, by using a truncation function, we define a new impact equations such that the dynamics of the solutions is simple in a neighbourhood of the origin so that we can avoid the arguments for complicated dynamical behaviour of the solutions produced by the forced term nearby the origin. Secondly, we will introduce a new coordinate transformation, transform the impact phase-plane from right half plane to the whole plane. Thus we can use the similar arguments in In Chapter 3 to construct a seies of topological quadrangles and apply the bend-twist theorm to obtain the existence of infinite periodic solutions. These periodic solutions has large norm, therefore we prove the existence of infinite periodic bouncing solutions for the original impact oscillators.
In Chapter 5, we discuss the existence of the global defined solutions and chaotic dynamics of solutions to a class of super-linear impact oscillators with indefinite weight. At first, doing as in Chapter 4, we introduce a new coordinate transformation to translate the impact systems into a new equal systems which defined in the whole phase-plane. Second, by using the qualitative method, we get the existence of solutions that defined in the intervals in which the weight change sign for finite times. And in this case, the solutions have different impact times given beforehand in the intervals where the weight take positive number or negative number. This fact help to obtain the existence of global defined solutions. Latter, by some delicate analysis for the dynamics of solutions in the phase plane, we prove that, for each infinite dimentional non-negative integer vector there is a global defined impact solution such that the impact times of the solution in each positive weight interval or negative weight interval, are the same as the number in the corresponding component. When the weight is periodic, above results imply that the impact oscillator exhibits chaotic-like dynamics.
引文
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