Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

We present two dual oscillating circuits having a wide spectrum of dynamical properties but relatively simple topologies. Each circuit has five bifurcating parameters, one nonlinear element of cubic current–voltage characteristics, one controlled element, LCR components and a constant biasing source. The circuits can be considered as two coupled oscillators (linear and nonlinear) that form dual jerk circuits. Bifurcation diagrams of the circuits show a rather surprising result that the bifurcation patterns are of the Farey sequence structure and the circuits’ dynamics is of a fractal type. The circuits’ fractal dimensions of the box counting (capacity) algorithm, Kaplan–Yorke (Lyapunov) type and its modified (improved) version are all estimated to be between 2.26 and 2.52. Our analysis is based on numerical calculations which confirm a close relationship of the circuits’ bifurcation patterns with those of the Ford circles and Stern–Brocot trees.