Math 2.0

Ordinary differential equations Qualitative theory Theorem

Poincaré–Bendixson theorem

Builds on Limit cyclePeriodic orbitEquilibrium pointFlow of a vector field

Statement

Let UU be an open subset of ℝ2\mathbb{R}^2, let f:U→ℝ2f : U \to \mathbb{R}^2 be continuously differentiable, and let φ\varphi be the flow of x′=f(x)x' = f(x). Let y∈Uy \in U be a point whose solution exists for all t≥0t \ge 0. Let ω(y)\omega(y) be its ω\omega-limit set, as in the definition of a limit cycle. Assume that ω(y)\omega(y) is non-empty and compact, and that it contains only finitely many equilibrium points. Then ω(y)\omega(y) is one of the following three kinds of set. (1) It is a single equilibrium point. (2) It is a periodic orbit. (3) It is a connected set of finitely many equilibrium points and of trajectories that tend to these points as t→∞t \to \infty and as t→−∞t \to -\infty.

Remarks

A common corollary: let K⊆UK \subseteq U be compact, contain no equilibrium point, and contain the positive half-trajectory of yy. Then ω(y)\omega(y) is a periodic orbit. To apply it, one often finds an annulus that trajectories enter but cannot leave. Then the annulus contains a periodic orbit. The theorem is special to the plane. It fails in ℝ3\mathbb{R}^3, where chaotic behaviour is possible, as for the Lorenz system. The trajectories in case (3) are homoclinic or heteroclinic orbits.

History

The theorem is named after Henri Poincaré and Ivar Bendixson.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary 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 levelTrajectorydefinitionPeriodic orbit, definition, no evidence levelPeriodic orbitdefinitionPhase plane, concept, no evidence levelPhase planeconceptLimit cycle, definition, no evidence levelLimit cycledefinitionPoincaré–Bendixson theorem, statement, no evidence levelPoincaré–Bendixson th…statement
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. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Order of a differential equation definition
  8. Solution of an ordinary differential equation definition
  9. Autonomous system definition
  10. Initial value problem definition
  11. Linear differential equation of order n definition
  12. Equilibrium point definition
  13. System of linear differential equations definition
  14. Uniqueness of solutions for Lipschitz vector fields statement
  15. Maximal solution definition
  16. Flow of a vector field definition
  17. Trajectory definition
  18. Periodic orbit definition
  19. Phase plane concept
  20. Limit cycle definition
  21. Poincaré–Bendixson theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem, Theorem
About this page
Tag0EW1X61ZT1
URIm2://0EW1X61ZT1
Kindstatement, theorem
WritingW1 stated
MSC34C05, 37C27
StatusThis copy of the site has no build data. Run m2 status 0EW1X61ZT1.
UsesLimit cycle, Periodic orbit, Equilibrium point, Flow of a vector field
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