trajektorioiden

Trajektorioiden are geometric objects used to describe families of trajectories in a dynamical system. The term is Finnish in origin, with trajektorioiden as the genitive plural of trajektorioide, and is sometimes rendered in English as “trajectoryoids.” In this sense, a trajektorioide encodes how individual trajectories vary with initial conditions or external parameters, rather than focusing on a single integral curve.

Definition and construction: Let M be a smooth manifold and X a vector field generating a flow φt. Let P be a parameter space and {γp : I → M} be the family of integral curves of X indexed by p ∈ P. The trajektorioide corresponding to this family is the subset Φ = { (γp(t), p) : p ∈ P, t ∈ I } ⊆ C∞(I,M) × P, or, more concretely, the image of the map F(p,t) = γp(t) together with the projection to P. This construction emphasizes the dependence on the parameter p.

Properties: Trajektorioiden capture the global behavior of a trajectory family and may carry a natural fibered or foliated structure when the curves fill space without transversal intersections. Under regularity assumptions, the trajektorioide can be given a smooth structure and studied using tools from foliation theory and jet spaces.

Examples: In a planar system with varying initial points, the trajektorioide records how trajectories bend as initial conditions change; in control theory, the family of response trajectories to different inputs forms a trajektorioide.

See also: trajectory, integral curve, dynamical system, foliation, jet bundle.