Dynamic Behavior Of Discrete Mapping

Chaos dynamics is an emerging discipline, originated in the early 20th century, formed in the 1960s. In the study of chaos dynamics theory, all kinds of chaos are constantly being found: the weather changes, neural network, the heart beating,and so on. Chaos exists in almost all fields of science, particularly in physics, mathematics, fluid mechanics, economics, biology, and so on. Chaos theory in information science, fluid mechanics, economics,biology, engineering and other fields has a very wide range of applications value. Therefore, the study of chaos dynamics and thus control of chaos has a very far-reaching and important significance!Chaos has four main features: 1) sensitivity to initial conditions; 2) elongation and folding; 3) there is strange attractors; 4) the rich level and self-similar structure. The feature of elongation and folding usually was showed by Baker transform in mathematics. In the paper, methods of the binary,ternary and M-Ary were used in studying respectively of the dynamic behavior of Baker transformation with the kinetic parameters of 2, 3,N. The initial value is taken, in the case of limited decimal number, repeating decimal, irrational number, the result is that tending to being restrained in the attractor, the periodic orbit, entering the chaotic state. Then the two-dimension Baker transformation was extended to three-dimension space and studied its dynamic characteristics and illustrated the Baker transform in image encryption Finally, dynamic characteristics of tent map with the kinetic parameters of 2 were studied by the method of binary. Pointing out that the result is tending to 0, being restrained in the periodic orbit, entering the chaotic state when the initial value are limited decimal number, repeating decimal, no repeating decimal. Dynamic characteristics of tent map with the kinetic parameters of 3 dynamics were studied by the method of ternary, the initial value was expressed as ternary decimals, if the ternary decimals do not contain 1, and its dynamic characteristics are the same as results of the binary. If the ternary decimals contain 1, and one is not the last one, then the result will be the escape set; if ternary decimal contains 1, and only the last decimal places of the decimal, the final will converge to 0.In the end, the relationship of the tent map and the Cantor was discussed, indicating "there was random in certainty" and "there was certainty in random"...