The Complex Dynamics of Large Systems with Long-Range Interactions: New Insights from Covariant Lyapunov Vectors

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In many real-world systems of interest, disorder is generated locally, leading to complex dynamics as a result of short and long-range spatial interactions with neighboring regions. Examples include the dynamics of the atmosphere and oceans, the patterns of chemicals in industrial processes, the dynamics of large distributed networks such as the internet, and the nonlinear interactions of large numbers of neurons. We quantify the impact of these spatial interactions on the overall dynamics using two model systems that contain the essential physics while remaining computationally accessible. We use the powerful idea of covariant Lyapunov vectors, which quantify the growth or decay of small disturbances, to build a deeper understanding of the dynamics of large systems with spatial interactions. These findings will provide insight for the development of the theoretical ideas needed to describe complex dynamical systems of societal interest. The project includes the development of a hands-on numerical workshop for junior high students, provides opportunities for undergraduate research, and supports the graduate research of a PhD student. The research findings, and the state-of-the-art computational approaches explored, will be used as part of an advanced graduate course.This project is a fundamental numerical investigation of the dynamics of large spatially-extended systems, with a range of spatial interactions, that are strongly driven out of equilibrium. We focus on two model systems that contain the essential nonlinearities and spatial interactions, while remaining computationally accessible, for a fundamental and broad study using powerful ideas from dynamical systems theory. We will explore large lattices of discrete-time maps and a canonical pattern forming partial differential equation called the Generalized Swift-Hohenberg equation. We will use the powerful idea of covariant Lyapunov vectors (CLV's) to gain fundamental new insights. The CLV's will yield an unprecedented and quantitative description of the tangent-space dynamics. We will quantify the degree of hyperbolicity of the dynamics, estimate the dimension of the inertial manifold, explore the generalization and robustness of the tangent-space splitting into physical and transient modes, and quantitatively link the pattern dynamics with the spatiotemporal dynamics of the CLV's. We will build a physical understanding for how these findings vary as a function of the strength and length-scale of the spatial interactions that are included in the model systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.