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本文用数值和分析的办法讨论了:(1)一维映象中的Hausdorff维数与Ляпунов指数的关系d(λ),指出在N(ε)关系中存在反映点集在不同尺度上的结构的多标度区现象,认真分析这个现象对d的计算有重要意义;(2)根据动力系统Poincar映象与一维映象的关系,及一维映象d(λ)在λ=0处的不连续性和多值性,猜测动力系统λ=(0,0,—)处的d(λ)关系也具有不连续性和多值性;(3)在d(λ)关系的基础上,以λ>0为判据,用统计抽样法估计了logistic模型混沌区内混沌轨道在参数轴上的测度,结果为mc=0.893±0.022(归一于混沌区的总测度为1)。In this paper, we use numerical and analytical methods to discuss the following problems. (1) The first is the relation between Hausdorff dimension d and Lyapunov exponent λ. We point out that in the relation N(ε) there is a multi-scaling-region phenomena which shows the structure of the point set at different scale. To analyse this phenomena seriously is significant for calculating d. (2) Acorrding to the relation between the one-dimensional map and the Poincare map of dynamic system, and the discontinuity and multivalue property of d( λ ) of one-dimensional map at λ=0, we conjecture that the relation d(λ) at λ= (0,0,-) of the dynamical system is also discontinuous and multivalued. (3) On the base of relation d(λ), we use λ>0 as a criterion to estimate the measure of chaotic solutions, of logistic model in chaotic region, on the axis of parameter. The result is mc= 0.893±0.022 (The total measure of chaotic region is normalized to unity).
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