Dynamics MOC

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:

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

Collectively trajectories are described by , where is the initial condition and is , , or . These have the so-called semigroup property

the name of which becomes more clear when written using the Time evolution operator

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 .²

Continuity of trajectories

If is a differentiable vector field, then for each there is an and such that has solutions for and where and is continuous in and ³.

Iterated map

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

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