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Dynamical systems and ergodic theory Definition

Intermittency

Builds on Bifurcation theoryPeriodic orbitSaddle-node bifurcationHopf bifurcationPeriod-doubling bifurcationLogistic map

Definition

Let (ϕμ)(\phi_\mu) be a one-parameter family of dynamical systems, flows or maps, with μ∈ℝ\mu \in \R. Assume that for μ\mu on one side of μ0\mu_0 the system has a stable periodic orbit, and that it disappears or loses stability at μ0\mu_0. The system shows intermittency for μ\mu just past μ0\mu_0 if typical orbits alternate between two kinds of phases.

  1. Laminar phases: long stretches of time during which the orbit stays close to the former periodic orbit.
  2. Bursts: short irregular stretches, after which the orbit is reinjected near the former periodic orbit.

The type is I when the bifurcation at μ0\mu_0 is a saddle-node bifurcation. It is II for a subcritical Hopf or Neimark–Sacker bifurcation, and III for a subcritical period-doubling bifurcation.

Remarks

Example: the logistic map x↦rx(1−x)x \mapsto r x (1 - x) near rc=1+22r_c = 1 + 2\sqrt{2}. At rcr_c a stable and an unstable orbit of period 33 are born in a saddle-node bifurcation of the third iterate. For rr slightly below rcr_c, orbits pass slowly through the narrow gaps left between the graph of f3f^3 and the diagonal, so they look nearly periodic of period 33. Between these passages they move chaotically: this is type I intermittency. For type I the mean length of the laminar phases grows like (rc−r)−1/2(r_c - r)^{-1/2}. Non-example: just after a supercritical period-doubling bifurcation the new stable orbit of double period attracts nearby orbits, so there are no bursts. Intermittency is a route to chaos studied in bifurcation theory.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Bifurcation theory, concept, no evidence levelBifurcation theoryconceptBorel 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 levelFunctionconceptHopf bifurcation, concept, no evidence levelHopf bifurcationconceptInterval, 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…conceptPeriodic orbit, concept, no evidence levelPeriodic orbitconceptSaddle-node bifurcation, concept, no evidence levelSaddle-node bifurcati…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 mapexampleIntermittency, definition, no evidence levelIntermittencydefinition
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. Bifurcation theory concept
  2. Borel sigma-algebra concept
  3. Differential equation concept
  4. Discrete-time dynamical system concept
  5. Euclidean space object
  6. Fixed point concept
  7. Function concept
  8. Hopf bifurcation concept
  9. Interval concept
  10. Invariant measure concept
  11. Lyapunov exponent concept
  12. Outer measure concept
  13. Period-doubling bifurcation concept
  14. Periodic orbit concept
  15. Saddle-node bifurcation concept
  16. Topological conjugacy concept
  17. Topological entropy concept
  18. Ordinary differential equation definition
  19. Lebesgue measure object
  20. Solution of an ordinary differential equation definition
  21. Tent map example
  22. Initial value problem definition
  23. Exponential growth and decay example
  24. Logistic equation example
  25. Logistic map example
  26. Intermittency definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Intermittency
About this page
TagF3STQVV22B
URIm2://F3STQVV22B
Kinddefinition
WritingW0 stub
MSC37G35, 37D45
DefinesIntermittency
UsesBifurcation theory, Periodic orbit, Saddle-node bifurcation, Hopf bifurcation, Period-doubling bifurcation, Logistic map

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