Badard R
Institut National des Sciences Appliquees de Lyon, Dept. Informatique, INSA-Lyon, 69621 Villeurbanne Cedex, France.
Chaos. 2008 Jun;18(2):023127. doi: 10.1063/1.2937016.
Iterations on R given by quasiperiodic displacement are closely linked with the quasiperiodic forcing of an oscillator. We begin by recalling how these problems are related. It enables us to predict the possibility of appearance of strange nonchaotic attractors (SNAs) for simple increasing maps of the real line with quasiperiodic displacement. Chaos is not possible in this case (Lyapounov exponents cannot be positive). Studying this model of iterations on R for larger variations, beyond critical values where it is no longer invertible, we can get chaotic motions. In this situation we can get a lot of strange attractors because we are able to smoothly adjust the value of the Lyapounov exponent. The SNAs obtained can be viewed as the result of pasting pieces of trajectories, some of which having positive local Lyapounov exponents and others having negative ones. This leads us to think that the distinction between these SNAs and chaotic attractors is rather weak.
由准周期位移给出的实数集(R)上的迭代与振荡器的准周期强迫密切相关。我们首先回顾一下这些问题是如何关联的。这使我们能够预测对于具有准周期位移的实直线简单递增映射出现奇异非混沌吸引子（SNA）的可能性。在这种情况下不可能出现混沌（李雅普诺夫指数不可能为正）。研究实数集(R)上这种迭代模型在更大变化范围内，即在其不再可逆的临界值之外的情况，我们可以得到混沌运动。在这种情况下，我们可以得到许多奇异吸引子，因为我们能够平滑地调整李雅普诺夫指数的值。所得到的奇异非混沌吸引子可以看作是粘贴轨迹片段的结果，其中一些片段具有正的局部李雅普诺夫指数，而另一些具有负的局部李雅普诺夫指数。这使我们认为这些奇异非混沌吸引子与混沌吸引子之间的区别相当微弱。
