Optimal periodic orbits of continuous time chaotic systems
Optimal periodic orbits of continuous time chaotic systems
复制标题
连续时间混沌系统的最优周期轨道
DOI：
10.1103/physreve.62.1950
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Ott
中科院分区：
文献类型：
--
作者：
Yang;Hunt;Ott
在以前的工作[B. R. Hunt和E. Ott，Phys. Rev. Lett. 76，2254（1996）; Phys.Rev.E54，328，（1996）]，基于数值实验和分析，证明了从混沌吸引子上所有可能的轨道中选出的最优轨道“典型地”是低周期的周期轨道。通过最优轨道，我们指的是产生系统状态的给定平滑“性能”函数的时间平均值的最大值的轨道。因此，最优性是相对于给定的性能函数定义的。(The对最优轨道的研究至少在三个方面是有意义的：控制混沌，高维动力系统的低维吸引子嵌入低维测量空间，以及同步混沌系统的冒泡分叉。在这里，我们扩展了以前的工作。特别是，之前的工作是针对离散时间动力系统的，这里我们将考虑连续时间系统（流）。流的一个本质区别是，混沌吸引子可以嵌入其中，不仅是不稳定的周期轨道，而且是不稳定的稳态，我们发现，最优性往往可以发生在稳态。我们还进一步阐明了最优性“通常”在低时期实现的意义。特别是，我们发现，作为一个系统参数被调整到更接近混沌吸引子的危机，最优性可能会出现在较高的时期。
In previous work [B. R. Hunt and E. Ott, Phys. Rev. Lett. 76, 2254 (1996); Phys. Rev. E 54, 328, (1996)], based on numerical experiments and analysis, it was conjectured that the optimal orbit selected from all possible orbits on a chaotic attractor is "typically" a periodic orbit of low period. By an optimal orbit we mean the orbit that yields the largest value of a time average of a given smooth "performance" function of the system state. Thus optimality is defined with respect to the given performance function. (The study of optimal orbits is of interest in at least three contexts: controlling chaos, embedding of low-dimensional attractors of high-dimensional dynamical systems in low-dimensional measurement spaces, and bubbling bifurcations of synchronized chaotic systems.) Here we extend this previous work. In particular, the previous work was for discrete time dynamical systems, and here we shall consider continuous time systems (flows). An essential difference for flows is that chaotic attractors can have embedded within them, not only unstable periodic orbits, but also unstable steady states, and we find that optimality can often occur on steady states. We also shed further light on the sense in which optimality is "typically" achieved at low period. In particular, we find that, as a system parameter is tuned to be closer to a crisis of the chaotic attractor, optimality may occur at higher period.