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Lawrence Virgin : Identifying chaos using spectral content



The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent (LE) spectrum establishes a quantitative measure, with at least one positive LE (and generally bounded motion) indicating a local exponential divergence of adjacent trajectories. Other measures are associated with certain geometric features of a chaotic attractor, e.g., the fractal dimension, and broadband frequency content. However, although the extraction of LE's can be accomplished with (necessarily noisy) experimental data, this is still a relatively data-intensive, sensitive (and frustrating) endeavor.
We present here an alternative, pragmatic approach to identifying chaos as a function of system parameters, based on frequency content and extending the concept of the spectrogram. This talk will describe this approach applied to systems of increasing complexity, ranging from direct numerical simulations of familiar archetypal systems like Lorenz and the pendulum to experimental data generated from mechanical systems. The accuracy and utility of the approach, including the effect of noise, is tested relative to the standard (LE) approach.

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