A Third-Order Differential Equation with Chaotic Solutions

The solution of the following simple third-order ordinary differential equation exhibits chaos [1]:

;

here is the only free parameter. Despite the simplicity of the equation, its solutions are capable of producing a variety of dynamical behaviors, depending on the choice of the nonlinear function , which can depend on all three variables in the system: , , and . Three cases are considered here:

[1] A. S. Elwakil, K. N. Salama, and M. P. Kennedy, "An Equation for Generating Chaos and Its Monolithic Implementation," International Journal of Bifurcation and Chaos, 12(12), 2002 pp. 2885–2895. doi:10.1142/S0218127402006205.