Trajectories in a basin of attraction

Question

For all trajectories in a basin of attraction approaching and ending in a fixed point, are these trajectories one and the same?

Thanks in advance.

Answer 1

No, not all trajectories that lead to a specific point must be the same, regardless of whether this point is the fixed point of a basin of attraction or not.

Note first of all the difference between a trajectory and an orbit. A trajectory is the set of all future states starting from a specific initial state (in some engineering fields, it will be considered as the set of states over a finite timespan). An orbit is the path through the phase-space as related by the dynamics function of the system. The difference is that an orbit stretches to both $+\infty$ and $-\infty$ time, so a trajectory is generally the subset of an orbit.

From this, even for a 1-dimensional system, it is obvious that not all trajectories leading to a fix-point will be the same, although they may all lie on the same orbit.

Furthermore, for a 2- or higher-dimensional system, the trajectories are in any case not all the same. Think of a simple damped pendulum: there are symmetric solutions leading to the fixed-point (which is a straightforward example of orbits that do not overlap but lead to the same fixed-point), as well as various orbits that are infinitesimally close but never touch (different orbits by definition do not touch). A simple simulation (or you can google some graphs, i.e. here) should be all you need to convince yourself.

Answer 2

There can be many different trajectories that approach and end at a fixed point.

Take for example $\dot{x}=-x$ on the real line. In this case, there are only two different trajectories (right or left of the fixed point $x=0$). Both trajectories will approach and end at the fixed point.