Math 2.0

Dynamical systems and ergodic theory Definition

Crisis

Builds on Bifurcation theoryAttractorStrange attractorPeriodic orbitLogistic mapPeriod-doubling bifurcation

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 near μ0\mu_0 the system has a chaotic attractor AμA_\mu. A crisis at μ0\mu_0 is a bifurcation at which AμA_\mu changes discontinuously as μ\mu crosses μ0\mu_0. The attractor is suddenly destroyed or suddenly jumps in size. At a crisis, Aμ0A_{\mu_0} touches an unstable periodic orbit or its stable manifold. The crisis is a boundary crisis if this orbit lies on the boundary of the basin of AμA_\mu. It is an interior crisis if this orbit lies inside the basin.

Remarks

Example: the logistic map fr(x)=rx(1−x)f_r(x) = r x (1 - x) at r=4r = 4. For r=4r = 4, the interval [0,1][0, 1] is invariant and carries a chaotic attractor. The attractor contains the repelling fixed point 00 and its preimage 11, which lie on the boundary of the basin. For r>4r > 4, points near 1/21/2 are mapped above 11 and then escape to −∞-\infty, so the attractor is destroyed: a boundary crisis. After the crisis, typical orbits stay near the old attractor for a long time before escaping, a chaotic transient. Non-example: a period-doubling bifurcation of a periodic attractor is not a crisis, since the attractor changes continuously and is not chaotic. Crises are a basic route by which strange attractors appear or vanish 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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Attractor, concept, no evidence levelAttractorconceptBifurcation 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 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…conceptPeriodic orbit, concept, no evidence levelPeriodic orbitconceptStrange attractor, concept, no evidence levelStrange attractorconceptTopological 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 mapexampleCrisis, definition, no evidence levelCrisisdefinition
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. Attractor concept
  2. Bifurcation theory concept
  3. Borel sigma-algebra concept
  4. Differential equation concept
  5. Discrete-time dynamical system concept
  6. Euclidean space object
  7. Fixed point concept
  8. Function 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. Strange attractor 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. Crisis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Crisis_(dynamical_systems)
About this page
TagA2MX36J5GD
URIm2://A2MX36J5GD
Kinddefinition
WritingW0 stub
MSC37G35, 37D45
DefinesCrisis
UsesBifurcation theory, Attractor, Strange attractor, Periodic orbit, Logistic map, Period-doubling bifurcation

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