| Ever since Li TianYan and J.A. Yorke give the word of "chaos" in1975, chaos began to be an important subject of theoretical research. Over the past two decades, the study of chaos theory is more widely to penetrate the various areas of applications, and shows good prospects for application in many disciplines (such as communications, biology, mathematics, etc.). Chaos control and anticontrol are an important branch of the applications of chaos. In this paper, firstly we introduce the course of developing of chaos. Then we discuss wether the the product of two chaotic maps (in the sense of Devaney) is still chaotic and extended to a finite number of mappings. For any given linear time-invariant system, wether we can turn the system into chaotic in the sense of Li and Yorke.In this paper, we study on the above two questions:1. Firstly, we give a counrerexample to show that two-product maps, who are chaotic in the sense of Devaney, are not chaotic in the general topological spaces. Then we have a reaserch on the two conditions of Devaney chaos. Firstly, we remark that if maps possess dense periodic points, so does their product. But the product of topologically transitive maps may not be topologically transitive. Finally, we introduce the concept of topologically mixing in the general topological spaces and prove that if the product of two chaotic maps (or finite), which one of them is topologically mixing, is chaotic in the sense of Devaney.2. According to the redefining Marotto theory in2005, the problem about the chaos anti-control of a stable linear time-invariant system is studied. For any given stable LTI system, we design a simple nonlinear feedback controller with an arbitrarily small maximum magnitude by introducing a new simple sawtooth function. The controlled system becomes chaotic in the sense of Li and Yorke. Finally, an arbitrary stable LTI system is employed for the simulation verification. The results verify and visualize the theory and method developed in the paper. |
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