Period-Doubling Bifurcations and Associated Universal Properties Including Parameter Dependence

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Abstract
A theory of the period-doubling phenomenon of one-dimensional mappings of the form xn+1 = F(xn, r) is presented, which enables us to evaluate the continuous parameter (r) dependence of various quantities in the neighborhood of the accumulation point of bifurcations. It is based on the following functional equations: which are derived under a convergence assumption from a recursion relation between functions Gk(z, y) introduced by scaling F(2k)(x, r) with respect to both x and r. We transform the above equations into the form , where , and solve this new equation invoking a systematic approximation scheme based on δ−1y expansion of the right-hand side. The results, the convergence rate of bifurcation points δ and (and hence G(z, y)), are shown to be in good agreement with those of numerical simulations.

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