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Dynamical systems and ergodic theory Definition

Bifurcation theory

Builds on Dynamical systemOrdinary differential equationPhase portraitEquilibrium pointStructural stabilitySaddle-node bifurcation

Definition

Let Λ⊆ℝk\Lambda \subseteq \R^k be an open set of parameters. For each μ∈Λ\mu \in \Lambda, let x˙=F(x,μ)\dot x = F(x, \mu) be an ordinary differential equation on an open set of ℝn\R^n, with FF smooth in (x,μ)(x, \mu). Two such systems are topologically equivalent if a homeomorphism maps the orbits of one onto the orbits of the other. The homeomorphism must preserve the direction of time. A parameter value μ0∈Λ\mu_0 \in \Lambda is a bifurcation value if the following holds. Every neighbourhood of μ0\mu_0 contains some μ\mu whose system is not topologically equivalent to the system at μ0\mu_0. Bifurcation theory studies the bifurcation values of such families and the changes of the phase portrait that occur at them.

Remarks

The same definitions apply to families of maps x↦fμ(x)x \mapsto f_\mu(x), with topological conjugacy in place of equivalence. Example: for x˙=μ−x2\dot x = \mu - x^2 on ℝ\R, the value μ0=0\mu_0 = 0 is a bifurcation value. For μ<0\mu < 0 there is no equilibrium, for μ=0\mu = 0 there is one, and for μ>0\mu > 0 there are two, at ±μ\pm\sqrt{\mu}. This is a saddle-node bifurcation. Non-example: μ0=1\mu_0 = 1 is not a bifurcation value of this family. For all μ\mu near 11 there are two equilibria, μ\sqrt{\mu} attracting and −μ-\sqrt{\mu} repelling, and the phase portraits are equivalent. A system whose topological type is unchanged by every small perturbation of the vector field, not only by changes of μ\mu, is structurally stable. Local bifurcations concern a neighbourhood of an equilibrium or periodic orbit; global ones, such as homoclinic bifurcations, involve whole orbits.

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To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptDynamical system, concept, no evidence levelDynamical systemconceptEquilibrium point, concept, no evidence levelEquilibrium pointconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptSaddle-node bifurcation, concept, no evidence levelSaddle-node bifurcati…conceptStructural stability, concept, no evidence levelStructural stabilityconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinitionFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionTrajectory, definition, no evidence levelTrajectorydefinitionPhase plane, concept, no evidence levelPhase planeconceptPhase portrait, concept, no evidence levelPhase portraitconceptBifurcation theory, definition, no evidence levelBifurcation theorydefinition
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. Differential equation concept
  2. Dynamical system concept
  3. Equilibrium point concept
  4. Function concept
  5. Interval concept
  6. Ordinary differential equation concept
  7. Saddle-node bifurcation concept
  8. Structural stability concept
  9. Ordinary differential equation definition
  10. Lipschitz condition definition
  11. Grönwall's inequality statement
  12. Order of a differential equation definition
  13. Solution of an ordinary differential equation definition
  14. Autonomous system definition
  15. Initial value problem definition
  16. Linear differential equation of order n definition
  17. Equilibrium point definition
  18. System of linear differential equations definition
  19. Uniqueness of solutions for Lipschitz vector fields statement
  20. Maximal solution definition
  21. Flow of a vector field definition
  22. Trajectory definition
  23. Phase plane concept
  24. Phase portrait concept
  25. Bifurcation theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bifurcation_theory
About this page
TagPTRKN3DFAA
URIm2://PTRKN3DFAA
Kinddefinition
WritingW0 stub
MSC37G10, 34C23
DefinesBifurcation theory
UsesDynamical system, Ordinary differential equation, Phase portrait, Equilibrium point, Structural stability, Saddle-node bifurcation

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