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Bryan J. Glaz : The Quasi-Periodic Intermittency Attractor and a Fractional Calculus Perspective of Generalized Spectral Decompositions for Nonlinear Dynamical Systems



Complex, nonlinear dynamical systems are pervasive across many Army relevant scientific disciplines, including turbulence, robotics, and reconfigurable active matter. Typically, our engineering objectives are to control the dynamics. However, our inability to achieve these objectives for a variety of high-dimensional dynamical systems is due to a lack of mathematical tools to describe underlying salient, particularly when it comes to control of such systems. This talk focuses on the Koopman theoretic based spectral decomposition perspective for extracting these features.
In the first part of the talk, we demonstrate how external forcing (e.g. control) can be accounted for in the form of a parametric term added to the Koopman operator of the unforced system. By using the simple fluid dynamics example of streamwise oscillating flow over a cylinder, we establish an analogy with parametrically excited Hopf bifurcations. Quite unexpectedly, we happened upon a peculiar phenomenon that is not quite chaotic, not quite quasi-periodic, and not quite intermittent. We establish the theoretical underpinnings for this phenomenon, that we name Quasi-periodic Intermittency and we discuss the implications across a variety of physical systems. In the second part of the talk, a fractional calculus perspective of complexity will be introduced to generalize Koopman based spectral decompositions to systems with memory as a first step toward dealing with truly complex systems. The analysis leads to modes whose temporal behavior is anomalous and lacks a characteristic scale. The approach we propose may uncover inherent memory effects that would otherwise be obscured by conventional spectral methods.
The talk will conclude with a discussion of new follow-on research thrusts ranging from control for adaptive, reflexive robotic mobility to self-organized adaptation and reconfiguration of active matter systems. In addition, opportunities for students (undergraduate and graduate) and faculty to engage with the U.S. Army Research Laboratory will be discussed.

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