Perpetual points: New tool for localization of co-existing attractors in dynamical systems

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.