I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve.
Given a dynamical system $\dot{\textbf{x}}=F(\textbf{x})$ in $n$ dimensions, and an initial condition $\textbf{x}(t)=\textbf{0}$
Would the trajectory be a mapping from time to the phase plane? $T:\mathbb{R}\mapsto \mathbb{R}^n$?
Would the trajectory be a set of points on the phase plane?
Would the trajectory be a set of points in $\mathbb{R}^{n+1}$, containing both time $t$ and the vector $\textbf{x}(t)$?
Also is the word trajectory a synonym of solution curve? how do they differ?
Best Answer
The trajectory is a subset of the phase space -- which often is $\mathbb{R}^n$, but, depending on the dynamical system, this can be a suitably chosen manifold. A solution to the dynamical system is a mapping $\phi: \mathbb{R}_{>0} \to \mathbb{R}^n$, $\phi: t \mapsto \phi(t)$. The solution curve this mapping traces out in $\mathbb{R}^n$ is called the orbit or trajectory. This is a (almost always) one-dimensional geometric object, parametrised by the one-dimensional time parameter $t$, with a degree of smoothness governed by the smoothness of the dynamical system (in particular, its right hand side).
Degenerate examples of 'orbits', i.e. equilibria, are zero-dimensional; apart from those, all orbits are diffeomorphic to the entire real line $\mathbb{R}$, the half line $(0,\infty)$ (for example, orbits converging to a stable equilibrium) or an open interval $(a,b)$ (for example, heteroclinic orbits connecting two equilibria).
In particular, an orbit or trajectory does not have a directionality: the time reversed version of the original dynamical system would give rise to orbits which are identical to the 'original' ones, while the 'direction of travel' is reversed.