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Statements

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파이겐바움 함수 Feigenbaum function
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concept in dynamical systems
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When , we have a single intersection, with slope exactly , indicating that it is about to undergo a period-doubling. Relationship between and when . The fixed point becomes unstable, splitting into a periodic-2 stable cycle. Relationship between and when . Before the period doubling bifurcation occurs. The orbit converges to the fixed point . When , there are three intersection points, with the middle one unstable, and the two others stable. When , there are infinitely many intersections, and we have arrived at chaos via the period-doubling route. When , there are three intersection points, with the middle one unstable, and the two others having slope exactly , indicating that it is about to undergo another period-doubling. Relationship between and when . The tangent slope at the fixed point . is exactly 1, and a period doubling bifurcation occurs.
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ロジスティック写像2回反復グラフの周期倍化分岐の様(a = 3).png Logistic iterates with r=3.56994567.svg Logistic iterates r=3.45.svg Logistic iterates 3.4.svg Logistic map iterates, r=3.0.svg ロジスティック写像2回反復グラフの周期倍化分岐の様(a = 3.3).png ロジスティック写像2回反復グラフの周期倍化分岐の様(a = 2.7).png
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Feigenbaum Function
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FeigenbaumFunction
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