UNIVERSAL METRIC PROPERTIES OF NON-LINEAR TRANSFORMATIONS
UNIVERSAL METRIC PROPERTIES OF NON-LINEAR TRANSFORMATIONS
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DOI：
10.1007/bf01107909
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发表时间：
1979-01-01
影响因子：
1.6
通讯作者：
FEIGENBAUM, MJ
中科院分区：
文献类型：
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作者：
FEIGENBAUM, MJ
形式化地发展了函数方程在描述独立于特定f的高度分叉吸引子xn+1=λf(Xn)的精确局部结构中的作用。存在通用函数的层次结构sGR(X)，每个函数描述相同的局部结构，但在2rpoint的簇的级别上。−1(X)=−αgr(gr(x/α)，其中g=limr→∞greisting，−αg(X)=αg(g(x/α))是一个方程，它的解决定了g和α。∼g−δ−rh*式中，δ>；1和hare被确定为算子ℒ的相关特征值和特征向量：$$\Mathcal{L}\Left[\psi\Right]=-\Alpha\Left[{\psi\Left({g\Left({{x\Mathord{\Left/{\vphantom{x\Alpha}}\Right.\kern-\nulldelimiterspace}\Alpha}}\Right)}\Right)+g‘\Left({g\Left({{x\mathord{\Left/{\vphantom{x\Alpha}}\Right.\kern-\nulldelimiterspace}\Alpha}}\Right)}\Right)\psi\Left({-x}\mathord{\Left/{\vphantom{{-x}\Alpha}}\Right。我们猜想ℒ有一个超过1的唯一本征值，并且证明了这个δ就是λ收敛速度。然后形式(*)继续到所有的λ，而不仅仅是离散的λ随机分支值，Λ在这样的λ下的随机动力学被确定。这些结果适用于任何基本周期的高分叉。在ℒ的S谱猜想的假设下，我们进一步分析了高迭代λf‘s的极限的稳定性，从而在局部意义上建立了我们的理论.在此过程中，我们证明了高度迭代的λf’s是共轭的togr‘s，从而为获得一个选择的λf’s提供了一些基本的近似方案.
The role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1=λf(xn) independent of a specificfis formally developed. A hierarchy of universal functionsgr(x)exists, each descriptive of the same local structure but at levels of a cluster of 2rpoints. The hierarchy obeysgr−1(x)=−αgr(gr(x/α), withg=limr → ∞grexisting and obeyingg(x)= −αg(g(x/α), an equation whose solution determines bothgandα. Forrasymptoticgr∼ g − δ−rh*where δ > 1 andhare determined as the associated eigenvalue and eigenvector of the operator ℒ: $$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right]$$ We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλrather than just discreteλrand bifurcation valuesΛrand dynamics at suchλis determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of thegrlimit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate togr's, thereby providing some elementary approximation schemes for obtainingλrfor a chosenf.