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Nonlinear dynamical systems are notoriously difficult to analyze, since closed-form solutions of the equations of motion are generally not available. The most well known example of this difficulty is found in deterministic chaos, in which systems actively produce unpredictable and random-appearing behaviors, despite the simplicity of the governing equations. Equally important, though, are the complex changes in behaviors (bifurcations) these systems undergo when varying system control parameters. Despite the inherent intractability of nonlinear behavior, these days many fields are well acquainted with these kinds of complex behavior; they are found in a wide range of phenomena from fluid turbulence and electronic circuits to ecological and financial market dynamics. As a result of the pressing need to understand these highly complex behaviors a sophisticated and rich set of topological and geometric techniques have been developed to circumvent directly and analytically solving the equations of motion. Although formal and abstract, most of the mathematical results reduce to statements about structures in a system’s high-dimensional state space. Until recently, the difficulty of research and training in nonlinear dynamical systems has been in attempting “visualize” these structures. Over the last decade or so many researchers in the area of dynamical systems have implemented exploratory tools for probing nonlinear dynamical systems and for visualizing the underlying mechanisms. However, it has been only in the last several years with the development of very high performance graphics processors and innovative visualization tools, such as sensory-immersive cave environments, that a direct approach to interactive visualization of nonlinear dynamical systems has become possible. In an attempt to explore the benefits to research and teaching in nonlinear dynamical systems, we recently began a substantial effort to port our existing dynamical systems simulation and visualization tools to the UCD’s KeckCAVES environment. These consist of several components. The first is a basic simulation engine for numerically solving sets of ordinary differential equations and nonlinear discrete-time maps. The second is an interface between the simulation engine and visualization and interactive, real-time control of parameters and visualization modalities. Finally, there are a set of analysis tools that allow one to explore, for example, the degree of instability (Lyapunov characteristic exponents and mixing rates) and associated geometry (local stable and unstable manifolds, basins of attraction, and the entanglement of global stable and unstable manifolds). We have ported a sufficient number of the existing tools to have a proof of concept for the overall development of the planned dynamical systems tool suite. For example, though only a rough draft, they have already demonstrated the benefits of how an immersive environment gives one direct access to understanding complex geometric state space structures. In particular one of the most valuable lessons to date is the central role played by real-time feedback between the researcher and a running simulation. This is key to developing a sense of how a system is behaving and making sense of the mechanisms involved in generating complex behaviors. We have been pleasantly surprised at how much faster this learning process goes when using KeckCAVES. Within minutes of learning the basic interface users begin to focus in on the various geometric structures presented to them and to use the interactive controls to pose new questions and probe the simulated system further. Van Aalsburg, a physics graduate student currently working on earthquake dynamics, has done the initial implementation using Professor Crutchfield’s existing tools as a guide.
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