Trajectory

A trajectory is a solution to a Dynamical system and corresponds to a path in the Phase space intersecting the initial condition (i.e. initial state).¹ In principle, there are three kinds of trajectories:

A trajectory at a fixed point remains there for all time,
A Periodic solution consists of a finite number of points (discrete) or a closed curve (continuous)
All other trajectories are infinite and never intersect themselves

Collectively trajectories are described by $𝑞 : 𝑇 \times 𝑀 \to 𝑀 : (𝑡, 𝑥) \mapsto 𝑞 (𝑡, 𝑥)$ , where $𝑞 (0, 𝑥) = 𝑥$ is the initial condition and $𝑇$ is $ℝ$ , $ℤ$ , or $ℕ$ . These have the so-called semigroup property

𝑞 (𝑡 + 𝑠, 𝑥) = 𝑞 (𝑠, 𝑞 (𝑡, 𝑥))

the name of which becomes more clear when written using the Time evolution operator $Φ 𝑡 : ℝ 𝑛 \to ℝ 𝑛 : 𝑥 \mapsto 𝑞 (𝑡, 𝑥)$

Φ 𝑡 + 𝑠 = Φ 𝑠 \circ Φ 𝑡

Differential system

Since in principle any initial state is possible, every point in phase space belongs to a trajectory. Furthermore, since given an initial state the whole system evolves deterministically both forwards and backwards in time, no two trajectories may intersect in finite time. This depends on the Picard-Lindelöf theorem, which requires Lipschitz continuity of $𝑓$ .²

Continuity of trajectories

If $𝐹 : ℝ 𝑛 \to ℝ 𝑛$ is a differentiable vector field, then for each $𝑥 0 \in ℝ 𝑛$ there is an $𝜖 > 0$ and $𝑇 > 0$ such that $˙ 𝑥 = 𝐹 (𝑥)$ has solutions $𝑞 (𝑡, 𝑥)$ for $- 𝑇 < 𝑡 < 𝑇$ and $‖ 𝑥 - 𝑥 0 ‖ < 0$ where $𝑞 (0, 𝑥) = 𝑥$ and $𝑞 (𝑡, 𝑥)$ is continuous in $𝑡$ and $𝑥$ ³.

Iterated map

In the discrete case forward-time uniqueness is guaranteed, but reverse uniqueness only holds if $𝑓$ is invertible.

#state/develop | #lang/en | #SemBr

2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §1.2, p. 8 ↩
2022. Nonlinear Dynamics: A Concise Introduction Interlaced with Code ↩
See MATH3021 lecture 2. Citation needed. ↩

Trajectory

Differential system

Continuity of trajectories

Iterated map

Footnotes