  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.
