Perturbative Methods in Coupled Lattice Maps and Applications

使用微扰方法，本项目将研究称为耦合点阵映射的高维动力系统的性质。这些系统由给定的简单且易于理解的动力系统的许多副本组成，由晶格点索引，并通过局部耦合相互作用。该项目将主要关注解轨迹的统计和几何性质。将特别关注全局性质，包括吸引子的分形维数和李雅普诺夫指数以及相关的李雅普诺夫特征空间。特别令人感兴趣的是，在定义系统的参数（相变）的微小修改下，这种系统的统计特性可能不连续地变化。最后，在可能的情况下，对这些映射获得的结果将扩展到耦合流。brbrCoupled Lattice Maps是许多不同研究领域的有趣模型，包括统计力学、流体动力学、神经系统、食物链和微观经济学。在大多数这些领域中，兴趣在于轨迹的统计性质。这些性质可以解释相干宏观行为、时空混沌和相变的出现。这些模型最重要的特征是局部动力学的内在动力学不稳定性（混沌）。双曲（混沌）动力系统理论在低自由度系统中得到了很好的发展。它对多自由度系统的扩展提出了新的数学挑战，并在上述研究领域具有强大的应用潜力。

Using perturbative methods, this project will study properties of dynamical systems of very high dimension called Coupled Lattice Maps. These systems consist of many copies of a given simple and well-understood dynamical system, indexed by the points of a lattice, and interacting through a local coupling. The project will mainly focus on the statistical and geometric properties of solution trajectories. Particular attention will be devoted to global properties, including the fractal dimension of the attractor and the Lyapunov exponents and associated Lyapunov eigenspaces. Of particular interest is the possibility that the statistical properties of such systems may vary discontinuously under small modifications of the parameters that define the system (phase transitions). Finally, the results obtained for these maps will be extended, where possible, to coupled flows.brbrCoupled Lattice Maps are of interest as models in many different areas of research, including Statistical Mechanics, Fluid Dynamics, Neural Systems, Food Chains, and Microeconomics. In most of these fields the interest is in the statistical properties of the trajectories. These properties should explain the emergence of coherent macroscopic behavior, space-time chaos, and phase transitions. The most important characteristic of these models is the intrinsic dynamical instability (chaos) of the local dynamics. The theory of hyperbolic (chaotic) dynamical systems is well developed in systems with few degrees of freedom. Its extension to systems with many degrees of freedom presents new mathematical challenges and a strong potential for applications in the aforementioned fields of study.