Math 2.0

Dynamical systems and ergodic theory Definition

Trajectory

Builds on Autonomous systemFlow of a vector field

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. Let φ\varphi be the flow of the autonomous system x′=f(x)x' = f(x). Let x0∈Ux_0 \in U, and let I(x0)I(x_0) be the maximal interval of existence of the solution with x(0)=x0x(0) = x_0. The trajectory, or orbit, of x0x_0 is the set 𝒪(x0)={φt(x0):t∈I(x0)}⊆U.\mathcal{O}(x_0) = \{\varphi_t(x_0) : t \in I(x_0)\} \subseteq U. The positive half-trajectory is 𝒪+(x0)={φt(x0):t∈I(x0), t≥0}\mathcal{O}^+(x_0) = \{\varphi_t(x_0) : t \in I(x_0),\ t \ge 0\}, and the negative half-trajectory uses t≤0t \le 0.

Remarks

Two trajectories are either equal or disjoint. So the trajectories form a partition of UU. A trajectory is a single point exactly when x0x_0 is an equilibrium point. It is a closed curve when the solution is periodic and not constant. Some books use the word trajectory for the solution curve t↦φt(x0)t \mapsto \varphi_t(x_0) itself, and the word orbit for its image.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
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 levelTrajectorydefinition
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. 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

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orbit_(dynamics), Definition
About this page
Tag0BVT6VSB3S
URIm2://0BVT6VSB3S
Kinddefinition
WritingW0 stub
MSC37C10, 34C05
DefinesTrajectory
UsesAutonomous system, Flow of a vector field

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by