Sheet structure in global bifurcations of a driven [Formula Presented]-diode circuit

Sheet structure is found in a global bifurcation diagram of an [Formula Presented]-diode circuit driven by a sinusoidal voltage source [Formula Presented] [Formula Presented] Bifurcations of a driven [Formula Presented]-diode circuit have three interesting features: (1) The alternate appearance of large periodic windows and chaotic bands, where the period of each window increases exactly by one as [Formula Presented] is increased. (2) The repeated appearance of period-1 attractors and chaotic bands as [Formula Presented] is increased. (3) The existence of two different windows, each of period 2, 3, and 4. This paper attempts to provide a complete understanding of the global nature of the above features. Comprehending global bifurcations of systems, including chaotic behavior, naturally necessitates understanding the nature of stable and unstable periodic orbits, the latter being essential in most situations. The [Formula Presented]-diode circuit is no exception. This paper accomplishes such a task by (i) performing extensive measurements of bifurcations in the ([Formula Presented]) plane, (ii) simplifying the dynamics of the circuit without losing essential features of the observed bifurcations, and (iii) carefully analyzing the simplified dynamics from a global perspective. An analytical method in this paper is in (iii), where exact bifurcation equations are derived then the bifurcation diagrams are drawn in the ([Formula Presented], [Formula Presented], [Formula Presented]) space instead of on the ([Formula Presented]) plane. Here [Formula Presented] and [Formula Presented] are the frequency and the amplitude of the driving voltage source, and [Formula Presented] will be precisely defined. This three-dimensional picture reveals the properties of stable and unstable periodic orbits, and makes many of the global bifurcation mechanisms involved almost transparent. In particular, the following are found: (1) All the period-1 attractors and their associated unstable period-1 orbits constitute a sheet structure in the ([Formula Presented], [Formula Presented], [Formula Presented]) space, and hence belong to the same family. (2) Other periodic attractors of the same period and their associated unstable periodic orbits form a sheet structure in ([Formula Presented], [Formula Presented], [Formula Presented]) space, and therefore belong to the same respective families. A very good correspondence between the numerical and experimental results is obtained. The global structure revealed will also clarify the global bifurcation mechanisms of other systems, e.g., the gear meshing and the offshore compliant systems described by equations similar to the present system.