New Insight into Chaos: A Motion Weakly Attracted by Different Unstable Periodic Sequences

Chaos is considered to be essential for life, since it can bring diversity to the world. However, it is difficult to detect the pattern of chaos due to its apparent randomness. In this paper, a nonlinear system based on a Hénon-like map is studied to show the semi-ordered structure of its chaotic sequences. The periodic sequences of such a map are solved analytically, and the stability is analyzed. Multiple coexisting unstable sequences with different periods and the chaotic attractor are obtained. An evidence is found that chaos is not purely apparently random, and it is weakly attracted by different unstable periodic sequences. Due to the weak attraction, it transits from one unstable periodic sequence to another at random instances, and it appears unpredictable.