Math 2.0

Dynamical systems and ergodic theory Definition

Bifurcation diagram

Builds on Bifurcation theoryEquilibrium pointAttractorPitchfork bifurcationLogistic map

Definition

Let I⊆ℝI \subseteq \R be an interval of parameters and XX a state space. Let (ϕμ)μ∈I(\phi_\mu)_{\mu \in I} be a one-parameter family of dynamical systems on XX, flows or maps. For each μ\mu, let Aμ⊆XA_\mu \subseteq X be a chosen invariant set of ϕμ\phi_\mu. Typical choices are the set of equilibria and periodic points, or the attractor reached from a fixed initial point. The bifurcation diagram of the family for this choice is the set {(μ,x)∈I×X:x∈Aμ}.\set{(\mu, x) \in I \times X : x \in A_\mu}.

Remarks

Example: for x˙=μx−x3\dot x = \mu x - x^3 on ℝ\R, take AμA_\mu to be the set of equilibria. The equilibria solve x(μ−x2)=0x(\mu - x^2) = 0, so the diagram is the line x=0x = 0 together with the parabola μ=x2\mu = x^2. The branches meet at (0,0)(0, 0), the pitchfork bifurcation of bifurcation theory. Usually stable branches are drawn solid and unstable ones dashed. For the logistic map, one takes for AμA_\mu the limit points of the orbit of a typical initial point. Non-example: a plot of a single orbit xnx_n against the time nn, at one parameter value, is a time series, not a bifurcation diagram.

Reading path

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

Show the path as a graph
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…conceptEquilibrium point, concept, no evidence levelEquilibrium pointconceptEuclidean 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…conceptPitchfork bifurcation, concept, no evidence levelPitchfork bifurcationconceptTopological 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 mapexampleBifurcation diagram, definition, no evidence levelBifurcation diagramdefinition
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. Equilibrium point concept
  7. Euclidean space object
  8. Fixed point concept
  9. Function concept
  10. Interval concept
  11. Invariant measure concept
  12. Lyapunov exponent concept
  13. Outer measure concept
  14. Period-doubling bifurcation concept
  15. Pitchfork 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. Bifurcation diagram definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bifurcation_diagram
About this page
TagHS5ZEM7TWA
URIm2://HS5ZEM7TWA
Kinddefinition
WritingW0 stub
MSC37G35, 37G10
DefinesBifurcation diagram
UsesBifurcation theory, Equilibrium point, Attractor, Pitchfork bifurcation, Logistic map

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.