IJPAM: Volume 71, No. 2 (2011)
HIERARCHICAL SEQUENCES, BIFURCATIONS
AND FRACTAL PROPERTIES
College of Engineering and Information Sciences
630 US Highway 1, North Brunswick, NJ 08902, USA
Abstract. We present two dual oscillating circuits, each with five bifurcating parameters and one nonlinear element of cubic current-voltage characteristics. The circuits can be considered as two coupled oscillators (linear and nonlinear). 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. The circuits show a wide spectrum of interesting dynamical properties. 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.
Received: July 5, 2011
AMS Subject Classification: 47N70, 34C15, 37M20, 28A80
Key Words and Phrases: oscillating circuits, bifurcations, singularly perturbed systems, farey sequence, Stern-Brocot tree, Ford circles, fractals
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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395