Problems in Chaotic Dynamics

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Project Summary This proposal is for a three-year theoretical/computational program on the study of chaotic systems. The principal investigator and graduate students will carry out the work. It is anticipated that, through their participation in the proposed research, the graduate students will become proficient in the theory of nonlinear dynamical systems, the design and implementation of computer experiments, and the modeling of physical systems. Two problem areas will be addressed: 1) The onset of synchronization in systems consisting of many interconnected dynamical units: Synchronization in systems of many interconnected dynamical (perhaps chaotic) units depends on the characteristics of the individual connected units, on the topology of the coupling network, and on the strengths of the couplings along each link. Our past work in this area was on systems of chaotic units where the coupling was global (all-to-all) and all links had equal coupling strength, and focused on providing a general theory of the transition from incoherent behavior to periodic oscillation. We propose to greatly extend this work to much more general connection topologies. This problem is important in physical and chemical systems, but perhaps its greatest interest is for biological systems where coherent oscillations are extremely prevalent and apparently result from the interaction of many small units. 2) Chaotic mixing and advection in fluids: This general class of problems is extremely important for a large variety of applications, yet there remain very interesting and significant basic open problems. Our past work has introduced the concept of finite time Lyapunov exponents and large deviation theory to this area, and we have used this to study fractal dimension, power spectra, and structure functions in a variety of flow situations. The main area of our proposed investigation will be on the effect of rigid boundaries in confined chaotic flows (potentially significant in most laboratory experiments). Intellectual Merit The onset of synchronism in large highly connected systems is a basic problem of inherent interest, and addresses important real-world problems, such as oscillatory behavior in biological systems. Chaotic mixing and advection is a problem of fundamental importance with impact in fields ranging from atmospheric science to chemical engineering. For both problem areas, previous work by the PI will serve as a strong base and starting point for the proposed research. Broader Impacts The proposed activity will promote the training of graduate students in important areas of research, and, by interaction with the larger chaos group at the university, it will educate others in these issues. The understanding gained through this research will be useful in a variety of fields, including physics, biology , engineering, and meteorology.