Phase trajectory

The trajectory of a point in a phase space, representing how the state of a dynamical system changes with time. If the system is described by an autonomous system of ordinary differential equations (geometrically, by a vector field), then one speaks of the phase trajectory of the autonomous system (of the field), and one also uses this terminology when the solutions of the system are not defined for all values of $t$. The adjective "phase" is often omitted.

When the state is independent of $t$, the phase trajectory reduces to a point — an equilibrium position, and when the dependence on $t$ is periodic one obtains a closed phase trajectory (so one often speaks of a "periodic phase trajectory" ), which also includes the previous case (but, when speaking of a closed phase trajectory, one often means that it does not reduce to a point). Non-closed phase trajectories can, in general, be highly diverse; they are classified from various points of view in topological dynamics. A point $w$ of a non-closed phase trajectory divides it into two parts — the positive and negative semi-trajectories. They represent the states corresponding to $t\geq0$ and $t\leq0$, if the system has state $w$ at $t=0$. (The last definition also formally applies to a closed phase trajectory, but then both semi-trajectories coincide.)

Sometimes one means by a phase trajectory not simply a curve (as a set of points) or an oriented curve (on which a direction is distinguished corresponding to the change of state as $t$ increases), but a curve parametrized in the process of the motion along this curve of the phase point that arises in the system. This terminology is partly due to the fact that there is no generally accepted name for this parametrized curve. It is true that if a dynamical system is described by a system of differential equations, one speaks simply of solutions of the latter, but this terminology is not suitable in the general case, when a dynamical system is treated as a group of transformations $\{S_t\}$ of the phase space. (The function $t\to S_tw$ for fixed $w$ is sometimes called a "motion" , but in mathematics "motions" are usually transformations of the whole space.)

Comments

For some aspects of the classification of non-periodic trajectories, see Recurrent point.

A "picture" of "all" the phase trajectories in phase space is often referred to as a phase portrait of the dynamical system.

References

[a1]	N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 6, 14
[a2]	A.G. [A.G. Butkovskii] Butkovsky, "Phase portraits of control dynamical systems" , Kluwer (1990) (Translated from Russian)
[a3]	M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) pp. 190ff
[a4]	D.V. Anosov (ed.) V.I. Arnol'd (ed.) , Dynamical systems , I: Ordinary differential equations and smooth dynamical systems , Springer (1988) pp. 159ff (Translated from Russian)