Math 2.0

Dynamical systems and ergodic theory Low-dimensional dynamics Example

Logistic map

Builds on Discrete-time dynamical systemFixed pointPeriod-doubling bifurcationTent mapTopological conjugacyLogistic equation

Example

For a parameter r∈[0,4]r \in [0, 4], the logistic map is fr ⁣:[0,1]→[0,1],fr(x)=rx(1−x).f_r \colon [0, 1] \to [0, 1], \qquad f_r(x) = r x (1 - x). It is a discrete-time dynamical system on [0,1][0, 1]: the maximum of frf_r on [0,1][0, 1] is fr(1/2)=r/4≤1f_r(1/2) = r/4 \le 1 and fr≥0f_r \ge 0 there, so frf_r maps [0,1][0, 1] into itself.

Check of the first bifurcations. The fixed points are 00 and, for r>1r > 1, xr=1−1/rx_r = 1 - 1/r. Since fr′(x)=r(1−2x)f_r'(x) = r(1 - 2x), one gets fr′(0)=rf_r'(0) = r and fr′(xr)=2−rf_r'(x_r) = 2 - r. So for 1<r<31 < r < 3 the fixed point xrx_r attracts nearby points. At r=3r = 3 its derivative is −1-1, and the map has a period-doubling bifurcation. For rr slightly above 33 an attracting orbit of period 22 appears.

Remarks

As rr grows further, attracting orbits of periods 4,8,16,…4, 8, 16, \dots appear at parameters that accumulate at r∞≈3.5699r_\infty \approx 3.5699. For r=4r = 4 the map is topologically conjugate to the full tent map T(y)=1−∣1−2y∣T(y) = 1 - \abs{1 - 2y} by h(y)=sin⁡2(πy/2)h(y) = \sin^2(\pi y / 2). Indeed f4(h(y))=4sin⁡2(πy/2)cos⁡2(πy/2)=sin⁡2(πy)f_4(h(y)) = 4 \sin^2(\pi y/2) \cos^2(\pi y/2) = \sin^2(\pi y), and h(T(y))=sin⁡2(πy)h(T(y)) = \sin^2(\pi y) for y≤1/2y \le 1/2 and sin⁡2(π−πy)=sin⁡2(πy)\sin^2(\pi - \pi y) = \sin^2(\pi y) for y>1/2y > 1/2. For r=4r = 4 the measure with density 1πx(1−x)\frac{1}{\pi \sqrt{x(1 - x)}} on (0,1)(0, 1) is invariant. The map is the discrete-time analogue of the logistic equation of population growth.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptDifferential equation, concept, no evidence levelDifferential equationconceptDiscrete-time dynamical system, concept, no evidence levelDiscrete-time dynamic…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFixed point, concept, no evidence levelFixed pointconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptInvariant measure, concept, no evidence levelInvariant measureconceptLyapunov exponent, concept, no evidence levelLyapunov exponentconceptOuter measure, concept, no evidence levelOuter measureconceptPeriod-doubling bifurcation, concept, no evidence levelPeriod-doubling bifur…conceptTopological conjugacy, concept, no evidence levelTopological conjugacyconceptTopological entropy, concept, no evidence levelTopological entropyconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLebesgue measure, object, no evidence levelLebesgue measureobjectSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionTent map, example, no evidence levelTent mapexampleInitial value problem, definition, no evidence levelInitial value problemdefinitionExponential growth and decay, example, no evidence levelExponential growth an…exampleLogistic equation, example, no evidence levelLogistic equationexampleLogistic map, example, no evidence levelLogistic mapexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Borel sigma-algebra concept
  2. Differential equation concept
  3. Discrete-time dynamical system concept
  4. Euclidean space object
  5. Fixed point concept
  6. Function concept
  7. Interval concept
  8. Invariant measure concept
  9. Lyapunov exponent concept
  10. Outer measure concept
  11. Period-doubling bifurcation concept
  12. Topological conjugacy concept
  13. Topological entropy concept
  14. Ordinary differential equation definition
  15. Lebesgue measure object
  16. Solution of an ordinary differential equation definition
  17. Tent map example
  18. Initial value problem definition
  19. Exponential growth and decay example
  20. Logistic equation example
  21. Logistic map example

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Logistic_map
About this page
TagQN9X9BNKTX
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Kindexample
WritingW0 stub
MSC37E05, 37G15, 37N25
Instance ofDiscrete-time dynamical system
UsesDiscrete-time dynamical system, Fixed point, Period-doubling bifurcation, Tent map, Topological conjugacy, Logistic equation
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