International Symposium on Nonlinear Theory and its Applications
Accurate Formulas Locating Unstable Periodic Points in Chaos
Tetsushi Ueta, Kei Nagao,
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We characterize a fractal nature around the fixed point of a 2nd dimensional discrete system by using the directional coloring method. In the previous work, we developed a basic computation method to locate unstable periodic points. From a couple of computed points, we derive an approximated formulation giving an accurate location of any periodic point. But some corrections by Newton’s method are required to guarantee their numerical accuracy, especially for high-periodic points. In this paper, we consider an improvement of the formulation from careful observation of the data and an accurate fitting for some computed data. As a result, a vary accurate formulation is obtained, i.e., the formulation gives a very accurate location of the unstable periodic point corresponding to the given number of the period. It also indicate a exponential spiral, thereby a fractal nature of the system is formulated. Some numerical results are shown.