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- 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. (en)
- The relationship between and when r = 3. The tangent slope at the fixed point is exactly 1, and a period doubling bifurcation occurs. (en)
- When a is exactly 3, the graph touches the diagonal at exactly three points, resulting in three periodic points. (en)
- The relationship between and when r = 2.7, before the period doubling bifurcation occurs. The orbit converges to a fixed point . (en)
- Orbit diagram of the logistic map from r = 3.55 to r = 4 (en)
- Fixed point Example of monotonically decreasing convergence to (en)
- When a is slightly greater than 3, the graph passes the diagonal and splits into stable and unstable 3-periodic points. (en)
- 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 . (en)
- Relationship between and when . Before the period doubling bifurcation occurs. The orbit converges to the fixed point . (en)
- Relationship between and when . The fixed point becomes unstable, splitting into a periodic-2 stable cycle. (en)
- Magnification of the chaotic region of the map (en)
- 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. (en)
- 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. (en)
- Time series when r = 3.8282 (en)
- Time series when r = 3.828327 (en)
- When , there are infinitely many intersections, and we have arrived at chaos via the period-doubling route. (en)
- Spider diagram of the logistic map with parameter r = 4 and time series up to n = 500 for the initial value = 0.3. (en)
- When , there are three intersection points, with the middle one unstable, and the two others stable. (en)
- 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. (en)
- 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 (en)
- Fixed point Example of monotonically increasing convergence to (en)
- 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 . (en)
- When , we have a single intersection, with slope exactly , indicating that it is about to undergo a period-doubling. (en)
- The relationship between and when r = 3.3. becomes unstable and the orbit converges to the periodic points and . (en)
- Relationship between and when . The tangent slope at the fixed point . is exactly 1, and a period doubling bifurcation occurs. (en)
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