Prasad A, Ramaswamy R
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1999 Sep;60(3):2761-6. doi: 10.1103/physreve.60.2761.
We study the probability densities of finite-time or local Lyapunov exponents in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections, which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully developed chaos the probability density has a cusp. Exact results are presented for the logistic map, x-->4x(1-x). At intermittency the density is markedly asymmetric, while for "typical" chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.
我们研究低维混沌系统中有限时间或局部李雅普诺夫指数的概率密度。虽然多重分形形式描述了这些密度在渐近或长时间极限下的行为，但存在显著的有限尺寸修正，这些修正依赖于坐标。根据动力学状态的性质，局部李雅普诺夫指数的分布具有特征形状。对于间歇动力学以及在危机时刻，动力学相关性导致具有拉伸指数尾部的分布，而对于充分发展的混沌，概率密度有一个尖点。给出了逻辑斯谛映射(x→4x(1 - x))的精确结果。在间歇性情况下，密度明显不对称，而对于“典型”混沌，已知中心极限定理成立并得到高斯密度。局部分析提供了关于动力学吸引子上可预测性变化的信息。这些用于表征混沌吸引子上非均匀空间组织的密度对噪声具有鲁棒性，因此可以从实验数据中测量得到。
