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Nearly all nontrivial real-world systems are nonlinear dynamical systems.Chaos describes certain nonlinear dynamical systems that have a very sensitivedependence on initial conditions. Chaotic systems are always deterministic andmay be very simple, yet they produce completely unpredictable and divergentbehavior. Systems of nonlinear equations are difficult to solve analytically,and scientists have relied heavily on visual and qualitative approaches todiscover and analyze the dynamics of nonlinearity. Indeed, few fields havedrawn as heavily from visualization methods for their seminal innovations: fromstrange attractors, to bifurcation diagrams, to cobweb plots, to phase diagramsand embedding. Although the social sciences are increasingly studying thesetypes of systems, seminal concepts remain murky or loosely adopted. Thisarticle has three aims. First, it argues for several visualization methods tocritically analyze and understand the behavior of nonlinear dynamical systems.Second, it uses these visualizations to introduce the foundations of nonlineardynamics, chaos, fractals, self-similarity and the limits of prediction.Finally, it presents Pynamical, an open-source Python package to easilyvisualize and explore nonlinear dynamical systems' behavior.

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction