The Role of Period Tripling in the Development of a Self Similar Bifurcation Structure

We study theoretically the period tripling phenomenon and its role in the development of the bifurcation structure of the periodically forced Toda oscillator. The analytical studies reveal some interesting features in the context of previously known period tripling in one-parameter systems where the loss of stability of the original equilibrium used to be associated with the birth of a bifurcating saddle. In the case of Toda oscillator, a period-tripled saddlenode is born around the stable original period where the saddle approaches the original period as the dissipativity is reduced. In the conservative limit the saddle undergoes a tangential interaction with the original centre. The recurrent appearance of period tripling and doubling suggest a new self similar feature of the associated bifurcation structure. The revelation of the underlying period tripling phenomena also helps to sequentially characterize each period, cascade and substructure, and to make some qualitative predictions regarding their locations in the phase and parameter space. The period tripling may give birth to some "degenerate period n orbits" (i.e. a certain class of period n orbits with identical n). However, they are distinctly characterized on the basis of concept of period tripling and we also predict the "degeneracy" (the number of such period n orbits for a given n) analytically. The same concept is extended in the case of degenerate cascades and substructures.