Dynamics and Kinetics

动力学和动力学

• 批准号: 1265883
• 负责人: Leonid Bunimovich
• 金额: $ 18.3万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265883&HistoricalAwards=false
• 关键词: Dynamics; Kinetics

The proposed research addresses some fundamental problems in the theory of dynamical systems and develops further a new promising direction, pioneered by the PI, which is concerned with a finite time properties of dynamics. A classical approach to studies of chaotic dynamics deals only with the asymptotic in time properties. A new approach to the design of hyperbolic (chaotic) elastic impact systems will be developed which will allow to essentially extend this class. It is based on a new general characterization of absolutely focusing mirrors in terms of continued fractions. A 40-years old problem on smoothening of stadium elastic impact systems will be resolved, which will allow to advance in understanding dynamics of Hamiltonian systems with divided into regular islands and chaotic seas phase spaces. Dynamics of a finite size particles in nonconvex polygons will be shown to be chaotic on contrary to a general view that impact systems with point and finite size particles have similar dynamics. This will be applied to classical Ehrenfest's wind-tree model in statistical mechanics to demonstrate that this model may even surpass the richness of dynamics in the Lorentz gas. A new area of finite time qualitative properties of dynamical and stochastic systems will be advanced into studies of open systems and multidimensional systems. The proposed research will make an essential impact in science and technology. A new area of finite time qualitative properties of chaotic and of complex systems, pioneered by the PI, opens new promising avenues in the studies of transport in chaotic systems and dynamical networks. It can be, in particular, directly applied to real world networks in order to detect e.g. the most efficient receptors and transmitters among the elements nodes) and among the links of networks as well as to analysis and design of networks. A new approach to study closed systems by making "holes" at different places of their spaces of states and observing how position of such "hole" influences dynamics has already been taken up by physicists in numerical experiments and will be a useful and easily implemented tool in laboratory experiments as well in physics, chemistry and technology. Simple visual models of systems with impacts proposed by the PI will continue to be built as experimental devices in physics labs all over the world. As clear visual examples they play an important role in teaching nonlinear dynamics and complex systems to students and young scientists in all areas of science and engineering. The proposed research will have impact not only within the theory of dynamical systems but in other areas of Mathematics as well, especially in Mathematical Physics. Analysis of chaotic motion of finite size particles in nonconvex polygons may have applications for transport in nanotubes where particles propagate one after another and do not interact. The proposed research will enhance national and international collaborations with Mexico, Canada, Brasil, UK, Germany and Italy. Among the PI's collaborators are several females and Hispanics. Graduate and undergraduate students will be involved in the proposed research. It will also serve to a broad dissemination to enhance scientific and technological understanding via participation of the PI, his collaborators and students in interdisciplinary conferences as well as through special lectures for students and young researchers.

所提出的研究解决了动力系统理论中的一些基本问题，并进一步发展了由 PI 开创的新的有前途的方向，该方向关注动力学的有限时间特性。混沌动力学研究的经典方法仅涉及时间属性的渐近性。将开发一种新的双曲（混沌）弹性冲击系统设计方法，这将从根本上扩展此类系统。它基于绝对聚焦镜的连续分数的新一般特征。一个长达 40 年的体育场弹性冲击系统平滑问题将得到解决，这将有助于加深对分为规则岛屿和混沌海洋相空间的哈密顿系统动力学的理解。非凸多边形中的有限尺寸粒子的动力学将被证明是混沌的，这与具有点和有限尺寸粒子的撞击系统具有相似动力学的一般观点相反。这将应用于统计力学中经典的埃伦费斯特风树模型，以证明该模型甚至可能超越洛伦兹气体的动力学丰富性。动态和随机系统的有限时间定性特性的新领域将被推进到开放系统和多维系统的研究中。拟议的研究将对科学和技术产生重大影响。由 PI 开创的混沌和复杂系统的有限时间定性特性的新领域，为混沌系统和动态网络中的输运研究开辟了新的有希望的途径。特别是，它可以直接应用于现实世界网络，以检测例如元素节点之间和网络链路之间以及网络的分析和设计中最有效的接收器和发射器。通过在状态空间的不同位置制作“洞”并观察此类“洞”的位置如何影响动力学来研究封闭系统的新方法已经被物理学家在数值实验中采用，并且将成为实验室实验以及物理、化学和技术领域中有用且易于实施的工具。 PI 提出的具有影响的系统的简单视觉模型将继续作为世界各地物理实验室的实验设备进行构建。作为清晰的视觉示例，它们在向科学和工程各个领域的学生和年轻科学家教授非线性动力学和复杂系统方面发挥着重要作用。拟议的研究不仅会在动力系统理论中产生影响，而且还会在数学的其他领域，特别是数学物理领域产生影响。对非凸多边形中有限尺寸粒子的混沌运动的分析可能适用于纳米管中的传输，其中粒子一个接一个地传播并且不相互作用。拟议的研究将加强与墨西哥、加拿大、巴西、英国、德国和意大利的国内和国际合作。 PI 的合作者中有几位女性和西班牙裔。研究生和本科生将参与拟议的研究。 它还将通过PI、他的合作者和学生参加跨学科会议以及为学生和年轻研究人员举办特别讲座来广泛传播，以增强科学和技术理解。