| Research Abstract |
Chaotic phenomena are, roughly speaking, unpredictable ones. These subjects are examined by various simulations, but their theory is not easy to be used practically. Recently Ott, Grebogi and Yorke proposed the chaos control, which has a possibility of practical use. In this study, we tried to formulate the chaos control mathematically and to analyze it from both mathematical and practical points of view.
As for mathematical study, we considered chaos phenomena in term of infinitesimal behavior related to Lyapunov exponent, and studied fiberwise invariant measures, Ruelle invariants, lifts of protective flows, branch points of projectively Anosov flows, chain recurrent sets of protective flows, exceptional minimal sets of codimension two. By projectivizing the othogonal projection for the derivative of a flow, we obtain a flow of a fiber bundle whose fiber is homeomorphic to the projective space. This flow is called a projective flow. Lyapunov exponent represents the infinitesimal dilatation. The projective flow means the infinitesimal twist along the orbits. By these results, Ruelle invariant, the characteristic representing the infinitesimal twist, is valid for the determination of chaotic behavior as well as Lyapunov invariant. Ruelle invariant is easy to be treated in mathematical sense. Thus they seem to be valuable.
As for applications of mathematical theory to real phenomena, we formulated mathematically multi-agents systems, switching arrival systems, chaotic mills and atmospheric phenomena. For example, for chaotic mills, we derived Markus mills from them, and found that their chaotic behaviors come from Parry maps. Parry maps are mathematically defined and their chaos control can be naturally formulated. In this point of view, we stepped out for the theoretical analysis of the chaotic control.
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