Math 2.0

Dynamical systems and ergodic theory Definition

Poincaré map

Builds on Autonomous systemFlow of a vector fieldPeriodic orbit

Definition

Let UU be an open subset of ℝn\R^n with n≥2n \ge 2 and let f ⁣:U→ℝnf \colon U \to \R^n be continuously differentiable. Let φ\varphi be the flow of the autonomous system x′=f(x)x' = f(x). Let Γ\Gamma be a periodic orbit of period T>0T > 0 and let x0∈Γx_0 \in \Gamma. Let HH be a hyperplane through x0x_0 that does not contain the direction f(x0)f(x_0). A transversal section at x0x_0 is the intersection Σ\Sigma of HH with a small open ball around x0x_0. Then there are an open neighbourhood VV of x0x_0 in Σ\Sigma and a continuously differentiable function τ ⁣:V→(0,∞)\tau \colon V \to (0, \infty) with τ(x0)=T\tau(x_0) = T. For each x∈Vx \in V, the time τ(x)\tau(x) is the first time near TT at which the solution through xx returns to Σ\Sigma. The Poincaré map of Γ\Gamma at Σ\Sigma is P ⁣:V→Σ,P(x)=φτ(x)(x).P \colon V \to \Sigma, \qquad P(x) = \varphi_{\tau(x)}(x).

Remarks

The existence of τ\tau follows from the implicit function theorem, because f(x0)f(x_0) is not parallel to Σ\Sigma. The point x0x_0 is a fixed point of PP, and fixed points of PP correspond to periodic orbits near Γ\Gamma. If the eigenvalues of the derivative DP(x0)DP(x_0) all have absolute value less than 11, the orbit Γ\Gamma attracts nearby solutions. Example: in polar coordinates let r′=r(1−r2)r' = r(1 - r^2) and θ′=1\theta' = 1, with Γ\Gamma the unit circle and Σ\Sigma the positive xx-axis near (1,0)(1, 0). Every solution returns to Σ\Sigma after the time 2π2\pi, and solving the equation for rr gives P(r)=(1+(r−2−1)e−4π)−1/2P(r) = \left(1 + (r^{-2} - 1) e^{-4\pi}\right)^{-1/2}. Then P(1)=1P(1) = 1 and P′(1)=e−4π<1P'(1) = e^{-4\pi} < 1, so the circle is attracting. Non-example: in the same system, the vertical line x=1x = 1 near (1,0)(1, 0) does not give a transversal section at (1,0)(1, 0), since f(1,0)=(0,1)f(1, 0) = (0, 1) points along that line.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptPeriodic orbit, concept, no evidence levelPeriodic orbitconceptOrdinary 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 fielddefinitionPoincaré map, definition, no evidence levelPoincaré mapdefinition
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. Periodic orbit concept
  5. Ordinary differential equation definition
  6. Lipschitz condition definition
  7. Grönwall's inequality statement
  8. Solution of an ordinary differential equation definition
  9. Autonomous system definition
  10. Initial value problem definition
  11. Uniqueness of solutions for Lipschitz vector fields statement
  12. Maximal solution definition
  13. Flow of a vector field definition
  14. Poincaré map definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincaré_map
About this page
TagJPNZMT2ARS
URIm2://JPNZMT2ARS
Kinddefinition
WritingW0 stub
MSC37C27, 34C25
DefinesPoincaré map
UsesAutonomous system, Flow of a vector field, Periodic orbit
Changed2 November 2026

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