**Abstract:**The stable and unstable manifolds are invariant phase space surfaces

which physically describe the inflows and outflows of material close to the

unstable Lagrange equilibrium points in the restricted three body problem,

or the unstable Lagrange points at the end of the bar in rotating barred

galaxies. After a short general introduction to the topic, we will show how

the manifold inflows and outflows are of use in the interpretation of several

phenomena in a variety of scales of interest in dynamical astronomy. Examples

are i) the L4-L5 asymmetry of Jupiter Trojan asteroids, ii) the eccentricity

growth of navigation satellites under the sole action of natural forces, iii)

spiral structure in barred galaxies, and iv) galactic tidal streams.

**Bio:**Christos Efthymiopoulos is currently Associate Professor

at the Department of Mathematics Tullio Levi-Civita of the University of Padova.

Before moving to Italy, he served for 17 years (2003-2019) as research staff at

the Research Center for Astronomy and Applied Mathematics of the Academy of

Athens, in Greece, being Research Director since 2011. In the last three years he

has served as President of the Commission A4 (Celestial Mechanics and Dynamical

Astronomy) of the International Astronomical Union. His Research focuses on

applications of nonlinear dynamical systems in Dynamical Astronomy. He is

author of more than 80 articles in scientific journals, addressing a variety

of topics of current interest in dynamical astronomy, such as the dynamics

of Trojan asteroids, classical and general relativistic perturbation theory,

orbital and attitude dynamics of natural or artificial satellites, spiral

structure, and the tidal interactions of galaxies.