Math 2.0

Ordinary differential equations Definition

Periodic orbit

Builds on Autonomous systemTrajectorySolution of an ordinary differential equation

Definition

Let UU be an open subset of ℝn\mathbb{R}^n, let f:U→ℝnf : U \to \mathbb{R}^n be locally Lipschitz, and consider the autonomous system x′=f(x)x' = f(x). A solution γ:ℝ→U\gamma : \mathbb{R} \to U is periodic if it is not constant and there is a real number T>0T > 0 with γ(t+T)=γ(t)\gamma(t + T) = \gamma(t) for each t∈ℝt \in \mathbb{R}. The smallest such TT is the period of γ\gamma. A periodic orbit, or closed orbit, is the trajectory γ(ℝ)\gamma(\mathbb{R}) of a periodic solution γ\gamma.

Remarks

A non-constant periodic solution has a smallest period T>0T > 0, because γ\gamma is continuous and not constant. A periodic orbit is a simple closed curve in UU. Let a maximal solution γ\gamma that is not constant satisfy γ(t1+T)=γ(t1)\gamma(t_1 + T) = \gamma(t_1) for one t1t_1 and one T>0T > 0. Then, by uniqueness, γ\gamma is defined on ℝ\mathbb{R} and is periodic. For the harmonic oscillator x′=yx' = y, y′=−xy' = -x, each circle x2+y2=r2x^2 + y^2 = r^2 with r>0r > 0 is a periodic orbit of period 2π2\pi.

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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 inequalitystatementSolution 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 problemdefinitionUniqueness 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 orbitdefinition
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.
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  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. Solution of an ordinary differential equation definition
  8. Autonomous system definition
  9. Initial value problem definition
  10. Uniqueness of solutions for Lipschitz vector fields statement
  11. Maximal solution definition
  12. Flow of a vector field definition
  13. Trajectory definition
  14. Periodic orbit definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Periodic_point, Dynamical system
  2. Web https://en.wikipedia.org/wiki/Limit_cycle, Definition
About this page
Tag01DSNP0B4R
URIm2://01DSNP0B4R
Aliasesclosed-orbit, cycle
Kinddefinition
WritingW0 stub
MSC34C25, 37C27
DefinesPeriodic orbit
UsesAutonomous system, Trajectory, Solution of an ordinary differential equation

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