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一维单峰映象的拓扑熵

陈瑞熊 陈式刚

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一维单峰映象的拓扑熵

陈瑞熊, 陈式刚

THE TOPOLOGICAL ENTROPY OF ONE-DIMENSIONAL UNIMODAL MAPS

Chen Rui-xiong, Chen Shi-gang
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  • 在对Logistic映象数值计算的基础上,我们分析了一维单峰映象的逆轨道结构,证明了不同参数处逆轨道总数N(n)随求逆次数n而变化的递推公式。借此解析地求得了在倍周期区中h(f)≡0;在U序列RLR21的m=3+2l周期点上h(f)=logαmp,其中αmp为方程αm-2αm-2-1=0的最大实根;在2j-1常和2j带交界处hj(f)=(1/2)jlog2,由此可得聚点μ∞处拓扑熵的标度指数t=0.449806…。在此基础上,我们还求得了混沌区的周期窗口,U序列RLaRb所对应的各点处的拓扑熵,以及hR*Q(f)=(1/2)hQ(f)的关系。证明是在M.S.S.规则和“*”乘法则的基础上进行的。所以本文的结果对一维单峰映象是普适的。
    On the basis of computational calculation for the logistic map, we analyse the inverse orbital structure of one-dimensional unimodalmap and prove the different recurrence formulae which show that the total number of inverse orbit N(n) changes with inverting time n at different parameters. By this way, we analytically obtain h(f)=0 within the period-doubling region; h(f) = logamp on m = 3 + 21 period point of U -sequence RLR21, where amp is the largest real root of equation am-2am-2-1 = 0; hj(f) = (1/2)j log2 on the boundary between 2j-1 band and 2j band, from this result we find the scaling exponent of topological entropy near the accumulation point μ∞, t = 0.449806…. By means of above conclusion, we obtain the topological entropy within each "window" in chaotic region and on U-sequence RLaRb period points, we also get the relation of hR*Q(f)=(l/2)hQ(f). Because we carry out our proofs on the basis of M.S.S. rule and "*" composition law, the results in this paper are universal to all the one-dimensional unimodal maps.
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  • 文章访问数:  8271
  • PDF下载量:  589
  • 被引次数: 0
出版历程
  • 收稿日期:  1985-12-20
  • 刊出日期:  1986-05-05

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