Wang Xingang, Lai Ying-Cheng, Lai Choy Heng
Department of Physics, National University of Singapore, 117542 Singapore.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jul;74(1 Pt 2):016203. doi: 10.1103/PhysRevE.74.016203. Epub 2006 Jul 17.
Strange nonchaotic attractors (SNAs) were previously thought to arise exclusively in quasiperiodic dynamical systems. A recent study has revealed, however, that such attractors can be induced by noise in nonquasiperiodic discrete-time maps or in periodically driven flows. In particular, in a periodic window of such a system where a periodic attractor coexists with a chaotic saddle (nonattracting chaotic invariant set), none of the Lyapunov exponents of the asymptotic attractor is positive. Small random noise is incapable of causing characteristic changes in the Lyapunov spectrum, but it can make the attractor geometrically strange by dynamically connecting the original periodic attractor with the chaotic saddle. Here we present a detailed study of noise-induced SNAs and the characterization of their properties. Numerical calculations reveal that the fractal dimensions of noise-induced SNAs typically assume fractional values, in contrast to SNAs in quasiperiodically driven systems whose dimensions are integers. An interesting finding is that the fluctuations of the finite-time Lyapunov exponents away from their asymptotic values obey an exponential distribution, the generality of which we are able to establish by a theoretical analysis using random matrices. We suggest a possible experimental test. We expect noise-induced SNAs to be general.
奇异非混沌吸引子（SNA）此前被认为仅出现在准周期动力系统中。然而，最近的一项研究表明，此类吸引子可由非准周期离散时间映射或周期驱动流中的噪声诱导产生。特别是，在这样一个系统的周期窗口中，周期吸引子与混沌鞍点（非吸引混沌不变集）共存，渐近吸引子的李雅普诺夫指数均非正。小随机噪声无法引起李雅普诺夫谱的特征变化，但它可通过将原始周期吸引子与混沌鞍点动态连接，使吸引子在几何上变得奇异。在此，我们对噪声诱导的SNA及其性质特征进行了详细研究。数值计算表明，与准周期驱动系统中维度为整数的SNA不同，噪声诱导的SNA的分形维数通常取分数值。一个有趣的发现是，有限时间李雅普诺夫指数相对于其渐近值的波动服从指数分布，我们能够通过使用随机矩阵的理论分析来确定其普遍性。我们提出了一种可能的实验测试方法。我们预计噪声诱导的SNA是普遍存在的。
