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Dynamical systems and ergodic theory Definition

Feigenbaum–Cvitanović functional equation

Builds on FunctionFunctional equationFeigenbaum constants

Definition

The Feigenbaum–Cvitanović functional equation is the functional equation g(x)=−α g(g(x/α))for all x∈[−1,1],g(x) = -\alpha \, g\big(g(x/\alpha)\big) \quad \text{for all } x \in [-1, 1], in which the unknowns are a real number α>1\alpha > 1 and a function g ⁣:[−1,1]→[−1,1]g \colon [-1, 1] \to [-1, 1] such that:

  1. gg is even, that is, g(−x)=g(x)g(-x) = g(x) for all x∈[−1,1]x \in [-1, 1].
  2. g(0)=1g(0) = 1.
  3. gg is twice differentiable at 00 with g′′(0)<0g''(0) < 0, so that 00 is a quadratic maximum.

Remarks

Setting x=0x = 0 in the equation gives 1=−α g(1)1 = -\alpha \, g(1), so any solution has α=−1/g(1)\alpha = -1/g(1). Lanford proved, with computer help, that there is a real-analytic solution, and its constant is the Feigenbaum constant α=2.502907875…\alpha = 2.502907875\ldots. For that solution g(1)=−1/α≈−0.3995g(1) = -1/\alpha \approx -0.3995.

Non-example: a constant function g=cg = c does not solve the equation together with clause 2, since the equation gives c=−αcc = -\alpha c, so c=0c = 0, while clause 2 asks g(0)=1g(0) = 1.

The right side is T(g)T(g) for the renormalization, or period-doubling, operator T(g)(x)=−α g(g(x/α))T(g)(x) = -\alpha \, g(g(x/\alpha)). So the solution is a fixed point of TT. The other Feigenbaum constant δ≈4.6692\delta \approx 4.6692 is the only eigenvalue of modulus greater than 11 of the derivative of TT at this fixed point.

Reading path

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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 levelFunctionconceptFunctional equation, concept, no evidence levelFunctional equationconceptInterval, 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 mapexampleFeigenbaum constants, object, no evidence levelFeigenbaum constantsobjectFeigenbaum–Cvitanović functional equation, definition, no evidence levelFeigenbaum–Cvitanović…definition
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.
  • no evidence level
  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. Functional equation concept
  8. Interval concept
  9. Invariant measure concept
  10. Lyapunov exponent concept
  11. Outer measure concept
  12. Period-doubling bifurcation concept
  13. Topological conjugacy concept
  14. Topological entropy concept
  15. Ordinary differential equation definition
  16. Lebesgue measure object
  17. Solution of an ordinary differential equation definition
  18. Tent map example
  19. Initial value problem definition
  20. Exponential growth and decay example
  21. Logistic equation example
  22. Logistic map example
  23. Feigenbaum constants object
  24. Feigenbaum–Cvitanović functional equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Feigenbaum_constants
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MSC37E20, 39B12
DefinesFeigenbaum–Cvitanović functional equation
UsesFunction, Functional equation, Feigenbaum constants

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