3. Non-linear Systems

This chapter explores non-linear dynamics through simple mathematical models. Rather than solve or numerically integrate differential equations, a variety of graphical techniques in phase space are introduced to gain an intuitive and visual feel for the dynamics. Equilibrium points, velocity vectors, bifurcations, trajectories, and basins of attraction are introduced in one-dimensional systems. In two-dimensional systems, these concepts are expanded upon and nullclines and hysteresis are added, as well as the behavior of limit cycles and separatrices. Three-dimensional systems are explored in the context of chaotic and slow-fast systems. The ability for non-linear systems to display robustness and meta-stability are also discussed. Examples of non-linear dynamics are provided from weather, politics, evolutionary biology, music, art, and dance. More speculative applications include how the trajectories of attention may be a way to understand the nature of consciousness and how free will might exist in a deterministic world through chaotic unpredictability. The chapter concludes with questions for either reflection or group discussion as well as resources for further exploration.