Period-Doubling Bifurcations and Associated Universal Properties Including Parameter Dependence
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概要
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A theory of the period-doubling phenomenon of one-dimensional mappings of the form x_<n+1>=F(x_n,γ) 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: [numerical formula] which are derived under a convergence assumption from a recursion relation between functions G_k(z,y) introduced by scaling F^<(2k)>(x,r) with respect to both x and r. We transform the above equations into the form λ^^^(y)=Φ(λ^^^(1+δ^<-1>y)), where λ^^^(y)=(∂^iG(0,y)/∂z^i), and solve this new equation invoking a systematic approximation scheme based on δ^<-1>y expansion of the right-hand side. The results, the convergence rate of bifurcation points δ and λ^^^(y) (and hence G(z,y)), are shown to be in good agreement with those of numerical simulations.
- 理論物理学刊行会の論文
- 1982-06-25
著者
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DAIDO Hiroaki
Department of Physics, Kyoto University
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Daido Hiroaki
Research Institute For Fundamental Physics Kyoto University
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DAIDO Hiroaki
Research Institute for Fuudamental Physics, Kyoto University
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