Vallejo Juan C, Viana Ricardo L, Sanjuán Miguel A F
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):066204. doi: 10.1103/PhysRevE.78.066204. Epub 2008 Dec 4.
By using finite Lyapunov exponent distributions, we get insight into both the local and global properties of a dynamical flow, including its nonhyperbolic behavior. Several distributions of finite Lyapunov exponents have been computed in two prototypical four-dimensional phase-space Hamiltonian systems. They have been computed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals. We analyze how such distributions serve or not for tracing the orbit nature and local flow properties such as the unstable dimension variability, as the axes are allowed or not to tend to the largest stretching direction. The relationship between the largest and closest to zero exponent distribution is analyzed. It shows a linear dependency at short intervals, related to the number of degrees of freedom of the system. Finally, the hyperbolicity indexes, associated to the shadowing times, are calculated. They provide interesting information at very local scales, even when there are no Gaussian distributions and the values cannot be regarded as random variables.
通过使用有限李雅普诺夫指数分布，我们深入了解了动态流的局部和全局特性，包括其非双曲行为。在两个典型的四维相空间哈密顿系统中计算了几种有限李雅普诺夫指数分布。它们是通过计算在给定区间内任意指向的一组正交轴的增长率来计算的。我们分析了随着轴是否趋向于最大拉伸方向，这种分布如何用于追踪轨道性质和局部流特性，如不稳定维度变异性。分析了最大指数分布和最接近零的指数分布之间的关系。它在短区间内显示出线性相关性，这与系统的自由度数量有关。最后，计算了与跟踪时间相关的双曲性指数。即使不存在高斯分布且这些值不能被视为随机变量时，它们在非常局部的尺度上也提供了有趣的信息。
