Mixed-mode oscillations and chaotic solutions of jerk (Newtonian) equations

Abstract

We analyze jerk equations (third-order ODEs) resulting from an underlying prototypical model of mixed-mode oscillations and propose their circuit realizations in this paper. The scalar ODEs and their corresponding circuit realizations are obtained from a system of first-order ODEs with one nonlinearity (third-degree polynomial). One of the jerk equations is Newtonian as it is obtained by computing the time-derivative of the second Newton’s law x″−F/m=0 for a constant mass m and specially designed nonlinear force function F(x,x′,τ). The second jerk equation is non-Newtonian. The two circuits are op-amp RC circuits with interesting dynamical properties, including the mixed-mode and chaotic oscillations. The mixed-mode oscillations follow the rules of Farey arithmetic and the circuits’ dynamics is of a fractal nature.