Dynamics and Kinetics

这项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开创的一个新的有前途的方向，即关于动力学的有限时间性质。研究混沌动力学的经典方法只涉及时间性质的渐近性。将开发一种新的方法来设计双曲(混沌)弹性碰撞系统，这将允许从本质上扩展这类系统。它基于一个新的关于连分式绝对聚焦镜的一般刻画。解决了体育场弹性碰撞系统的光顺问题，使人们对哈密顿系统的动力学有了更深入的了解。哈密顿系统分为规则岛和混沌海区。有限大小粒子在非凸多边形中的动力学将被证明是混沌的，这与一般的观点相反，即具有点和有限大小粒子的碰撞系统具有相似的动力学。这将应用于统计力学中经典的Ehrenfest风树模型，以证明该模型甚至可能超过洛伦兹气体中的动力学丰富程度。动力学和随机系统的有限时间定性性质的一个新领域将被推进到开放系统和多维系统的研究中。拟议中的研究将对科学和技术产生重要影响。PI开创了混沌和复杂系统有限时间定性性质的新领域，为研究混沌系统和动态网络中的输运问题开辟了新的途径。特别是，它可以直接应用于真实世界网络，以便检测例如元素节点之间以及网络链路之间的最有效的接收器和发射器，以及用于网络的分析和设计。一种研究封闭系统的新方法，即在状态空间的不同位置打“洞”，并观察这种“洞”的位置如何影响动力学，已经被物理学家在数值实验中采用，并将在实验室实验中成为一种有用且易于实现的工具，在物理、化学和技术中也是如此。PI提出的具有碰撞的简单可视系统模型将继续作为实验设备在世界各地的物理实验室中建立。作为清晰的视觉范例，它们在向所有科学和工程领域的学生和青年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。这项拟议的研究不仅将对动力系统理论产生影响，而且还将对数学的其他领域产生影响，特别是在数学物理学方面。分析有限尺寸粒子在非凸多边形中的混沌运动可能在纳米管中的传输中有应用，在纳米管中，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、德国和意大利的国家和国际合作。少年派的合作者中有几名女性和西班牙裔。研究生和本科生将参与这项拟议的研究。它还将通过国际和平协会、其合作者和学生参加跨学科会议以及通过为学生和青年研究人员举办特别讲座来广泛传播，以增进对科学技术的了解。

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.