A Hybrid of Theoretical and Computational Methods for Bifurcation Analysis

本文主要研究复杂系统全局吸引子结构的动力学分岔现象，需要采用理论和严格计算相结合的方法。研究人员考虑了在近似下不具有鲁棒性的特定现象，例如出现无限数量的分岔，或者当相关参数增加时，偏微分方程系统在相空间收敛到一点的解。具体来说，该提案解决了分岔图中周期加倍级联的出现以及扩展系统中平衡簇的出现，并旨在开发用于建立异斜连接的严格计算工具。所开发的方法将应用于多组分合金中cahn - moral体系相分离过渡态的研究，以及Ohta Kawasaki双嵌段共聚物模型中复杂过渡模式的研究。为这些例子开发的方法具有普遍的意义。它们将作为解决类似棘手计算问题的原型。此外，为严格计算而开发的数值工具具有通用性和广泛的适用性。这项工作集中在理论和计算之间的边界，这样每一种方法单独失败，但理论和计算一起产生成功的结果。其中几个例子侧重于回答材料科学中的具体问题，包括合成化学中的软材料模型，例如用作粘合剂添加剂的嵌段共聚物，以及二元和多组分金属合金的相分离。这项研究有助于更好地理解和开发新材料。其中一个被研究的模型与大气科学直接相关，因为它被用来研究云中雨滴的形成。从长远来看，这些方法还将应用于化学和生物学中出现的各种模式形成问题。学生将从一个已经存在的本科生和研究生研究小组中加入这个项目。两位调查人员都致力于招募女性学生和代表性不足的少数民族成员。

This proposal focuses on dynamical bifurcation phenomena for which it is necessary to use a combination of theoretical and rigorous computational methods in order to study the global attractor structure for complicated systems. The investigators consider specific phenomena which are not robust under approximation, such as the occurrence of an infinite number of bifurcations, or solutions for systems of partial differential equations converging to one point in phase space as an associated parameter increases. Specifically, the proposal addresses the appearance of period-doubling cascades in bifurcation diagrams as well as the occurrence of equilibrium clusters in extended systems, and aims at the development of rigorous computational tools for establishing heteroclinic connections. The developed methodology will be applied to the study of transitional states in Cahn-Morral systems for phase separation in multicomponent alloys, and the study of complicated transition patterns in the Ohta Kawasaki diblock copolymer model. The methods developed for these examples are of general interest. They will serve as prototypes for solving similar intractable computational problems. In addition, the numerical tools developed for rigorous computation are general purpose and widely applicable.This work is focused at the boundary between theory and computation, such that each method fails individually, but theory and computation together yield successful results. Several of the examples focus on answering specific questions in materials science, including models for soft materials in synthetic chemistry such as block copolymers used as additive in adhesives, and phase separation for binary and multicomponent metal alloys. The research enables the better understanding and development of new materials. One of the studied models is immediately relevant in atmospheric science, as it is used to study the creation of raindrops within a cloud. In the longer term, there will be additional applications of these methods to a variety of pattern formation problems arising in chemistry and biology. Students will be incorporated into this project from within an already existing research group of undergraduate and graduate students. Both investigators are committed to recruiting students who are women and members of underrepresented minorities.