Basic Concepts of Chaos Theory and Nonlinear Time-Series Analysis

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