On Chaos of the Logistic Maps
نویسندگان
چکیده
This paper is concerned with chaos of a family of logistic maps. It is first proved that a regular and nondegenerate snap-back repeller implies chaos in the sense of both Devaney and Li-Yorke for a map in a metric space. Based on this result, it is shown that the logistic system is chaotic in the sense of both Devaney and Li-Yorke, and has uniformly positive Lyapunov exponents in an invariant set for a certain parameter interval with a lower bound less than a specific value, at which the unique 3-periodic orbit appears. In addition, it shows the exact parameter range for the existence of an asymptotically stable 3-periodic point, and consequently the exact parameter range for the biggest periodic window, i.e., 3-periodic window, in the period-doubling bifurcation diagram.
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