Period-doubling cascades in one and two parameter maps



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Period doubling cascades are one of the most prominent features of one-parameter families of maps, F : R x M → M, where M is a locally compact manifold without boundary, typically RN. The dissertation is divided into two parts. In the first part period doubling cascades in one parameter maps are considered. In the second part we consider the implications of period doubling cascades in maps with two parameters. Our first set of results builds on 60 years of history of period doubling cascades in one parameter families. It has been previously proved that under certain conditions on a family of maps, cascades persist even under large perturbations in one dimensional maps and in coupled quadratic maps. We extend this work to prove that the cascades persist if we couple N one dimensional maps which satisfy certain conditions with a coupling g. Then, we use this result to prove the existence of cascades for a coupled family of cubic maps and for a mixed family of quadratic and cubic maps. Based on an established method for enumerating the period of a cascade, we are able to enumerate the cascades for a series of concrete examples including degree five and degree six polynomial maps. In the second part, we concentrate on results for period doubling cascades in two parameter families of maps. Starting with the work of Milnor it has been observed that when a bifurcation diagram for a map with multiple extrema is drawn in the parameter space, it is seen to contain many shrimp-like structures. A shrimp in the two parameter space can be viewed as a counterpart of a stable periodic window in one parameter bifurcation diagram. Many different methods have been used to investigate the parameter space like isoperiodic diagrams, continuation methods and the Lyapunov graph method. We use information from the first part to conclude results for the previously discussed shrimps for the cubic map. We demonstrate that shrimps occur in degree five and degree six polynomial maps derived in the previous section. We show that the shrimps appear in the parameter space for the values of the parameters where stable periodic window appears in the bifurcation diagram. In the past shrimps have not been observed in the quadratic map. We also compare the bifurcation diagram and the Lyapunov graph of a population model with three parameters and show that typical shrimps do not exist in its parameter space. We conjecture that a typical unimodal map does not have shrimps with multiple legs.