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.