【作者】 黄国庆；

【导师】 伍歆；

【作者基本信息】 南昌大学 ， 固体力学， 2015， 博士

【摘要】 非线性动力学是非线性力学研究的一个重要领域。许多非线性保守和耗散动力系统具有对初始条件的极端敏感依赖性,即混沌现象。由于混沌系统的不可积性,无法得到其解析解。从理论和数值上对这些非线性系统的动力学研究涉及几个问题:如何寻找可靠的数值积分方法?怎样构造或采用可靠的混沌识别方法?如何了解非线性系统的动力学性质?如何运用非线性系统的动力学特性去解释一些物理现象?围绕这些问题,本文建立了以同一点两切向量夹角的余弦作为新混沌指标,并与已有的快速李雅普诺夫指标和较小排列指标等进行比较发现其一样具有较好的混沌识别灵敏性;还采用辛算法或Runge-Kutta(RK)为积分工具,研究了牛顿和1阶后牛顿圆型限制性三体保守系统动力学、圆轨道衰减的限制性三体耗散系统动力学及物理非线性弹性直杆件问题动力学。另外,利用高阶Runge-Kutta(RK)方法探讨了新四维自治耗散系统动力学,并用模拟硬件电路和基于单片机的数字电路实验进行仿真。下面分别简述这些工作。1、新混沌指标——余弦指标。考虑到快速Lyapunov指标与较小排列指标都是迅速识别混沌的指标,并且后者比前者识别混沌速度更快、更灵敏,因此,本文将以较小排列指标为基础发展和建立以同一点两切向量夹角的余弦作为相对论框架内独立于时空坐标选择并具有较好灵敏性的新混沌指标。如果轨道混沌,切空间同一点的两个切向量的夹角的余弦指数式地趋于1,而对于有序轨道,余弦一般在0与1之间某个值波动或代数式地趋于0。因此,余弦值可以作为区分有序和混沌轨道的指标。借助余弦指标和辛算法研究牛顿圆型限制性三体问题。使用几个辛算法分别求解该问题得到了它们的能量误差,找出精度最好算法;再用精度最好算法求解变分方程,即意味着全局辛算法的实施。结果表明新余弦混沌指标与Lyapunov指数和快速Lyapunov指标一样都可以正确揭示系统的有序和混沌性质,并且比Lyapunov指数识别混沌更快、更灵敏。2、1阶后牛顿圆型限制性三体问题动力学。对距离、时间和速度等标度变换得到质心旋转坐标系下的拉格朗日函数,使其中的牛顿圆型限制性三体问题部分两主天体距离与圆运动角速度都化为1,但后牛顿项明显含有两主天体距离与圆运动角速度的贡献。这样处理方便做牛顿与相对论的圆型限制性三体问题有序和混沌动力学的比较研究。通过大量扫描两主天体距离揭示1阶后牛顿三体问题轨道动力学定性演化规律。最后,由考虑两中心天体圆运动的1阶后牛顿效应的拉格朗日理论推导相应的1阶后牛顿哈密顿,揭示二者在有序和混沌动力学定性上存在一些差异。3、圆轨道衰减的限制性三体问题动力学。对圆轨道有引力耗散衰减的圆型限制性三体问题的运动方程进行位置、速度和加速度标度因子变换,通过大量扫描两主天体距离,发现与牛顿圆型限制性三体问题不一样,有圆轨道衰减的圆型限制性三体问题的轨道是不稳定的,换言之,两主天体的最终运动状态必然是并合,而小天体必然逃逸;系统保持牛顿动力学性质的时间与两主天体的距离有关。两主天体距离越短,第三个小天体逃逸就越早。两主天体间距愈大,牛顿三体问题动力学特性维持的时间愈长。4、非线性粘弹性杆件问题动力学。在工程应用中,分析弹性细杆强度、刚度和稳定具有十分重要的意义。该力学问题考虑的是一端固定而另一端受周期拉伸的二次和三次非线性Keilven-voigt粘弹性直杆动力学。首先应用Galerkin方法将无限维动力系统转化为单模态、双模态和三模态动力方程,进一步得到对应的Hamilton系统。其次,采用四阶辛算法、四阶力梯度辛算法、最优化四阶力梯度辛算法和含有三阶导数项的辛算法分别计算两类不同的轨道以便比较这些辛算法的能量精度来挑选精度最好算法。再次,利用Poincaré截面、Lyapunov指数、快速Lyapunov指标和功率谱等研究直杆单模态系统分别在参数激励和强迫激励作用下存在分岔、周期、准周期和混沌现象。最后,揭示不管是无强迫自由振动保守系统还是有阻尼参数激励的非自治耗散系统都可能存在周期、准周期和混沌性质。5、新四维电路系统动力学。从模拟电路推导出新的四维自治微分方程,对系统平衡点进行稳定性分析,再采用Lyapunov指数(LCE)、快速Lyapunov指标(FLI)和较小排列指标(SALI)等识别该系统的有序、混沌和稳定性。发现FLI和SALI在区分这个耗散系统的混沌性要比LCE快很多。还运用上述指标和分岔图找到系统从有序到混沌的参数临界值(r=14.6)以及系统由弱混沌跃迁为超混沌的参数临界值(r=35.7)。同时设计了抗干扰的基于单片机的数字电路和硬件模拟电路来演示混沌的实现。更多还原

【Abstract】 Nonlinear dynamics is an important field in nonlinear mechanics. For many nonlinear conservative and dissipative systems, their evolutions exhibit exponentially sensitive dependence on initial conditions, i.e. chaotic behaviors. Because of nonintegrability, any chaotic system has no analytical solution. In theoretical and numerical researches on the subject of chaos, there are four poblems. What numerical integrator is good? What chaos indicator is reliable? How are dynamical properties of a nonlinear system understood? How are these qualitative properties used to explain any physical phenomena? Along these main points, this thesis discusses the dynamics of several conservative and dissipative systems as follows.1. A new cosine chaos index. Based on the small alignment index, a cosine chaos index as a new chaos indicator is given. It is the absolute value of a cosine of an angle between two tangent vectors at the same point. If it exponentially tends to 1 for a stable orbit, then the orbit is chaotic; if it remains a constant between 0 and 1 or tends to zero in a power law, then the orbit is ordered. A global fourth-order symplectic integrator that is superior to the other fourth-order methods in the accuracy is applied to solve the equations of motion and the variational equations on the classical circular restricted three-body problem. As a result, it is a quicker and more sensitive indicator to detect chaos from order than the method of Lyapunov exponents or the fast Lyapunov indicators.2. Post-Newtonian circular restricted three-body problem. By using scaling transformations to time and distance, we give the first post-Newtonian equations of motion for a relativistic circular restricted three-body problem, where the Newtonian terms do not depend on the separation of a parent binary, though the post-Newtonian terms do. The post-Newtonian contributions consist of the relativistic effects between the two primaries(i.e. the relativistic influence of the circular orbital frequency of the two primaries and of the separation between the two primaries to a third body). When the former post-Newtonian contribution in the nonrelativistic system is considered, the post-Newtonian dynamics are qualitatively different from the Newtonian dynamics if the separation between the two primaries is not large enough. As the latter post-Newtonian contribution is considered, some trajectories become unstable. Through a scan of dependence of the dynamics on the separation between two primaries with fast Lyapunov indicators, the dynamics are classified into three domains for dynamically unstable, bounded regular, and bounded chaotic dynamics under the circumstance of the two post-Newtonian contributions included.3. A circular restricted three-body problem with an orbital decay. In this system, an orbital decay due to an effect of gravitational radiation reaction between the two primary bodies is considered but the direct effect of gravitational radiation on the test particle is neglected. We use time-scale and distance- transformations to Newtonian problems so that Newtonian systems with orbital decay will depend on separation between the primaries but systems without orbital decay will not depend on this separation. There is a freedom of choice of various starting separations of the parent binaries in such a decay dissipative system for a given regular or chaotic orbit in the Newtonian counterpart. Thus, it is helpful to provide some insight into chaotic behavior of the third body in the decay dissipative case. When a large initial separation of the primaries is considered, chaos that exists in the Newtonian problem can be kept for a long enough time scale of the dissipative evolution before coalescence of the primaries. Of course, the final state of the third object is an escape due to the orbital decay.4. Dynamics of nonlinear viscoelastic rod. In the application, the analysis of the strength, stiffness and stability of rod is of great significance. A physical quadratic and cubic nonlinear equation on Keilven-voigt viscoelastic rod with one end fixed and the other end having periodic stretches is considered. The Galerkin method is applied to transform an infinite dimensional dynamical system into dynamic equations of single, double and three degrees of freedom, and a nonautonomous Hamiltonian system with two degrees of freedom is also given. The best one of the fourth-order common symplectic algorithm, the fourth-order gradient symplectic method, the optimized fourth-order forces one and the gradient symplectic method containing the first-, second- and third-order derivative terms is selected. Meantime, the methods of Lyapunov exponents, fast Lyapunov index, power spectrum and Poincare section are also used. Bifurcations and periodic, quasi-periodic and chaotic orbits are shown clearly.5. A new four-dimensional chaotic circuit system. The nonlinear system is derived from a circuit setup. Dynamically qualitative properties of individual orbits in this system are observed on an oscilloscope. Meanwhile, with the help of numerical investigations regarding the computations of Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes for finding chaos, we study the Lyapunov stability of the system and the dynamics of the regula and the chaotic individual orbits. Comparisons of numerical results and experimental ones show that the observed results coincide basically with the computed ones to a great extent. In particular, the bifurcation, Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes show almost the same rules of transitivity to chaos on a parameter of the system. It is worth pointing out that when the parameter has a threshold value from order to chaos, and the chaos varies stronger and stronger, the parameter is smoothly changed from small to large. In addition, the calculating of Lyapunov exponents can result in another threshold value of the parameter from chaotic behaviors to hyperchaotic ones. At the same time, a digtial circuit based on AVR32 singlechip and the analog circuit of this new chaotic system has been realized for finding chaotic behaviors. The simulated results confirm that the chaotic system can be realized and is a novel chaotic system.更多还原