On a phase space  and over a time interval  , consider a non-autonomous dynamical system defined through the flow map , mapping initial conditions  into their position  for any time . If the flow map  is a diffeomorphism for any choice of , then for any smooth set  of initial conditions in , the set
is an invariant manifold in the extended phase space . Borrowing terminology from fluid dynamics, we refer to the evolving time slice  of the manifold  as a material surface (see Fig. 1). Since any choice of the initial condition set  yields an invariant manifold , invariant manifolds and their associated material surfaces are abundant and generally undistinguished in the extended phase space. Only few of them will act as cores of coherent trajectory patterns.