The relationship between and when r = 2.7, before the period doubling bifurcation occurs. The orbit converges to a fixed point .
ロジスティック写像、安定2周期軌道、クモの巣図.png
When a is slightly greater than 3, the graph passes the diagonal and splits into stable and unstable 3-periodic points.
パラメータ3.82のロジスティック写像における微差の初期値から出発する2つの軌道.png
Logistic map bifurcation diagram magnifications.png
Orbit diagram for parameter r from −2 to 4. The orbit diverges when the parameter a goes beyond this range, both on the negative and positive sides.
یک مدل ساده ریاضی است که غالبا برای توصیف رشد جمعیت های زیستی استفاده می شود
Time series when r = 3.828327
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Mappa logistica
Chaotic orbits of the logistic map when r = 3.82. The orange squares are orbits starting from , and the blue-green circles are orbits starting from .
Graph of when r is slightly less than 3. The graph is not tangent except at the fixed points, and there are no 3-periodic points.
Mapa logístic
Logistic map time evolution a=3.828327.png
Relationship between and when . The tangent slope at the fixed point . is exactly 1, and a period doubling bifurcation occurs.
متتالية لوجستية
カオス的振る舞いを示す単純な2次式写像
horizontal
Fixed point Example of monotonically decreasing convergence to
Suite logistique
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.
Logistic map cobweb plot a=1.8.png
Logistic iterates 3.4.svg
May 2025
パラメータ3.82のロジスティック写像におけるズレの進展.png
When , there are three intersection points, with the middle one unstable, and the two others stable.
The relationship between and when r = 3.3. becomes unstable and the orbit converges to the periodic points and .
ロジスティック写像、安定2周期軌道、時系列.png
The relationship between and when r = 3. The tangent slope at the fixed point is exactly 1, and a period doubling bifurcation occurs.
Simple polynomial map exhibiting chaotic behavior
Logistic iterates with r=3.56994567.svg
center
Relationship between and when . The fixed point becomes unstable, splitting into a periodic-2 stable cycle.
ロジスティック写像
Mapa logístico
Logistic map
Magnification of the chaotic region of the map
Time series when r = 3.8282
Логістичне відображення
Fixed point Example of monotonically increasing convergence to
When a is exactly 3, the graph touches the diagonal at exactly three points, resulting in three periodic points.
逻辑斯谛映射
suite récurrente découlant du modèle de croissance limitée de Verhulst
The intermittency that occurs just before The part where the three almost identical values continue periodically is a laminar, and the part where chaotic irregular changes occur is a burst.
полиномиальное отображение
Spider diagram and time series for a = 3.3. The orbit is attracted to a stable 2-periodic point.
Stable regions within the chaotic region, where a tangent bifurcation occurs at the boundary between the chaotic and periodic attractor, giving intermittent trajectories as described in the Pomeau–Manneville scenario
When , we have a single intersection, with slope exactly , indicating that it is about to undergo a period-doubling.
mathematisches demographisches Modell
로지스틱 사상
The trajectory starting from x 0 = 0.1234 and ˆx The difference in orbits starting from grows exponentially. The vertical axis is ,shown on a logarithmic scale.
Logistic map iterates, r=3.0.svg
Relationship between and when . Before the period doubling bifurcation occurs. The orbit converges to the fixed point .
Logistic map time evolution a=3.8282.png
Logistic iterates r=3.45.svg
Orbit diagram of the logistic map from r = 3.55 to r = 4
Aplicación logística
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
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
When , there are infinitely many intersections, and we have arrived at chaos via the period-doubling route.
mapa polinomial simples exibindo comportamento caótico
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
Band structure. Because the spacing rapidly decreases, it is not possible to show more than eight bands. The top and bottom lines, which contain the orbitals, are within the range of equation .
Odwzorowanie logistyczne
혼돈적 움직임을 보이는 2차식 사상
Logistic map cobweb plot a=1.2.png
Logistische Gleichung
ロジスティック写像3回反復グラフの接線分岐の様子 .png
Логистическое отображение
Spider diagram of the logistic map with parameter r = 4 and time series up to n = 500 for the initial value = 0.3.
Subsection Bifurcation Diagram Logistic Map.png