Abstract
We introduce the basic concepts and methods to formalize and analyze deterministic chaos, with links to fractal geometry. A chaotic dynamic is produced by several kinds of deterministic nonlinear systems. We introduce the class of discrete-time autonomous systems so that an output time series can directly represent data measurements in a real system. The two basic concepts defining chaos are that of attractor—a bounded subset of the state space attracting trajectories that originate in a larger region—and that of sensitivity to initial conditions—the exponential divergence of two nearby trajectories within the attractor. The latter is what makes chaotic dynamics unpredictable beyond a characteristic time scale. This is quantified by the well-known Lyapunov exponents, which measure the average exponential rates of divergence (if positive) or convergence (if negative) of a perturbation of a reference trajectory along independent directions. When a model is not available, an attractor can be estimated in the space of delayed outputs, that is, using a finite moving window on the data time series as state vector along the trajectory.
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Sangiorgio, M., Dercole, F., Guariso, G. (2021). Basic Concepts of Chaos Theory and Nonlinear Time-Series Analysis. In: Deep Learning in Multi-step Prediction of Chaotic Dynamics. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-030-94482-7_2
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DOI: https://doi.org/10.1007/978-3-030-94482-7_2
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