  Perpetual points (PPs) are special critical points for which the magnitude of
acceleration describing dynamics drops to zero, while the motion is still
possible (stationary points are excluded), e.g. considering the motion of the
particle in the potential field, at perpetual point it has zero acceleration
and non-zero velocity. We show that using PPs we can trace all the stable fixed
points in the system, and that the structure of trajectories leading from
former points to stable equilibria may be similar to orbits obtained from
unstable stationary points. Moreover, we argue that the concept of perpetual
points may be useful in tracing unexpected attractors (hidden or rare
attractors with small basins of attraction). We show potential applicability of
this approach by analysing several representative systems of physical
significance, including the damped oscillator, pendula and the Henon map. We
suggest that perpetual points may be a useful tool for localization of
co-existing attractors in dynamical systems.
