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 article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior.
Highlights
Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems
Chaotic systems are a type of nonlinear dynamical system that may contain very few interacting parts and may follow simple rules, but all have a very sensitive dependence on their initial conditions [1,2]
Foundational conceptsof of nonlinear dynamics, chaos, fractals, self-similarity and the limits of. It argued that information visualization is a key way to engage with these system concepts
Summary
Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems. A system is a set of interacting components that form a larger whole. Few fields have drawn as heavily from visualization as nonlinear dynamics and chaos have for their pivotal discoveries, from Lorenz’s first visualization of strange attractors [31], to May’s groundbreaking bifurcation diagrams [32], to phase diagrams for discerning higher-dimensional hidden structures in data [33] Such nonlinear analysis is useful, yet underutilized for exploring time series [34,35]. We investigate the difference between chaos and randomness before visualizing the famous butterfly effect and discussing its implications for scientific prediction All of these models and visualizations are developed in Python using Pynamical; for readability, we reserve the technical details of its functionality for Appendix A
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