Random Attractors Found Using Lyapunov Exponents By Paul Bourke, October 2001 Contributions from Philip Ham and Johan Bichel Lindegaard --- Overview This document explores chaotic attractors generated by iterating a two-dimensional nonlinear quadratic map. It demonstrates how random selections of parameters in the map can produce a variety of dynamical behaviors, including fixed points, periodic orbits, divergence, and chaos. Visual examples of the attractors are illustrated throughout. --- The Quadratic Map The system considered is: \[ \begin{cases} x{n+1} = a0 + a1 xn + a2 xn^2 + a3 xn yn + a4 yn + a5 yn^2 \\ y{n+1} = b0 + b1 xn + b2 xn^2 + b3 xn yn + b4 yn + b5 yn^2 \end{cases} \] Parameters \(an, bn\) are randomly selected within bounds (e.g., \(\pm 2\)). Iterations are performed over large numbers of steps (e.g., 100,000), discarding early steps to allow settling. The Lyapunov exponent is computed to determine the system's behavior. --- Lyapunov Exponents and Chaos The Lyapunov exponent (\(\lambda\)) measures the average rate at which nearby trajectories diverge or converge. Systems have as many Lyapunov exponents as their phase-space dimensions, usually focusing on the largest exponent. Interpretation of \(\lambda\): Positive \(\lambda\): System is chaotic and unstable; nearby trajectories diverge exponentially, indicating sensitive dependence on initial conditions but deterministic behavior. Negative \(\lambda\): System converges to a fixed point or a stable periodic orbit (dissipative). Zero \(\lambda\): System is neutrally stable, generally conservative and steady-state. Visual illustrations display attractors corresponding to each case. --- Types of System Behavior Fixed Point Convergence The system settles to a single value; easier to detect by monitoring distance between successive points than by Lyapunov exponent (which can tend to \(-\infty\)). Divergence to Infinity Common with wide parameter ranges (\(\pm 2\)), such cases are discarded as they cause numerical instability. Periodic Orbits Attractors with negative Lyapunov exponents associated with repeating patterns. Chaos Positive Lyapunov exponent attractors filling a region of the plane—these create visually interesting and complex fractal-like images. --- Generation and Statistics Around 98% of random parameter sets result in divergence. Approximately 1% lead to fixed point attractors. Roughly 0.5% produce periodic attractors. The attractors shown were identified by filtering parameter sets via computed Lyapunov exponents. The program used to generate these attractors is available as gen.c. --- Visual and Conceptual Notes Some visually appealing structures may not be strictly chaotic attractors; they may lack a single basin of attraction. Two-dimensional nonlinear systems often generate images perceived as three-dimensional structures due to their fractal complexity. --- Additional Content Included are several attractor images demonstrating various dynamics. An unrelated image of a Space Shuttle launch smoke ring is provided for visual interest. --- References Berge, Pomeau, Vidal, Order Within Chaos, Wiley, 1984. Crutchfield, Farmer, Packard, Chaos, Scientific American, 1986. Das, Roy, Applicability of Lyapunov Exponent in EEG data analysis. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989. Feigenbaum, Universal behaviour in Nonlinear Systems, Los Alamos Science, 1981. Peitgen, Jurgens, Saupe, Chaos and Fractals - New Frontiers of Science, Springer, 1992. --- Contributions by Dmytry Lavrov Additional attractor images contributed by Dmytry Lavrov showcase further explorations in chaotic maps. --- This summary captures the methodology for discovering chaotic attractors through Lyapunov