Math 2.0

Dynamical systems and ergodic theory Definition

Flow of a vector field

Builds on Autonomous systemUniqueness of solutions for Lipschitz vector fieldsMaximal solution

Definition

Let UU be an open subset of ℝn\mathbb{R}^n and let f:U→ℝnf : U \to \mathbb{R}^n be locally Lipschitz. For x0∈Ux_0 \in U, let γx0:I(x0)→U\gamma_{x_0} : I(x_0) \to U be the maximal solution of the autonomous system x′=f(x)x' = f(x) with x(0)=x0x(0) = x_0. Let Ω={(t,x0):x0∈U, t∈I(x0)}\Omega = \{(t, x_0) : x_0 \in U,\ t \in I(x_0)\}. The flow of ff is the map φ:Ω→U\varphi : \Omega \to U given by φ(t,x0)=γx0(t)\varphi(t, x_0) = \gamma_{x_0}(t). Another notation is φt(x0)=φ(t,x0)\varphi_t(x_0) = \varphi(t, x_0).

Remarks

The flow has the group properties φ0(x)=x\varphi_0(x) = x and φs(φt(x))=φs+t(x)\varphi_s(\varphi_t(x)) = \varphi_{s + t}(x), whenever both sides are defined. The second property follows from uniqueness and time translation. The set Ω\Omega is open and φ\varphi is continuous on Ω\Omega. If Ω=ℝ×U\Omega = \mathbb{R} \times U, the flow is complete. Then t↦φtt \mapsto \varphi_t is an action of the additive group ℝ\mathbb{R} on UU. For a linear system x′=Axx' = A x, the flow is φt(x)=etAx\varphi_t(x) = e^{tA} x, the matrix exponential.

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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 fielddefinition
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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

Where this leads

43 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Flow_(mathematics), Autonomous systems of ordinary differential equations
  2. Web https://en.wikipedia.org/wiki/Flow_(mathematics), Formal definition
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Kinddefinition
WritingW1 stated
MSC37C10
DefinesFlow of a vector field
UsesAutonomous system, Uniqueness of solutions for Lipschitz vector fields, Maximal solution

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