A Hybrid of Theoretical and Computational Methods for Bifurcation Analysis

为了研究复杂系统的全局吸引子结构，需要采用理论和严格计算相结合的方法来研究动力学分叉现象。研究人员考虑了在近似下不稳健的特定现象，例如，随着相关参数的增加，出现无穷多个分支，或者偏微分方程组的解收敛到相空间中的一点。具体地说，该提案解决了分叉图中倍周期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宿连接的严格计算工具。所开发的方法将用于研究Cahn-Morral体系的过渡态，用于多元合金的相分离，以及研究Ohta Kawasaki两嵌段共聚物模型中的复杂转变模式。为这些例子开发的方法是普遍感兴趣的。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上，使得每种方法都单独失败，但理论和计算一起产生了成功的结果。其中几个例子侧重于回答材料科学中的具体问题，包括合成化学中软材料的模型，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。这项研究使人们能够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产生。从长远来看，这些方法将在化学和生物学中出现的各种图案形成问题上得到更多的应用。学生将从已经存在的本科生和研究生研究小组中纳入到这个项目中。这两名调查人员都致力于招收女性学生和代表性不足的少数族裔成员。

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.