A Visual Introduction to Nonlinear Systems: Chaos, Fractals, and the Limits of Prediction

Abstract

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This paper has three aims. First it discusses several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second it introduces to an interdisciplinary audience the foundations of nonlinear dynamics, chaos, fractals, self-similarity, and the limits of prediction – arguing that information visualization is a key way to engage with these concepts. Finally it presents Pynamical, a tool to visualize and explore nonlinear dynamical systems' behavior.