  In this paper we apply the method of Lagrangian descriptors to explore the
geometrical structures in phase space that govern the dynamics of dissipative
systems. We demonstrate through many classical examples taken from the
nonlinear dynamics literature that this tool can provide valuable information
and insights to develop a more general and detailed understanding of the global
behavior and underlying geometry of these systems. In order to achieve this
goal, we analyze systems that display dynamical features such as hyperbolic
points with different expansion and contraction rates, limit cycles, slow
manifolds and strange attractors. Furthermore, we study how this technique can
be used to detect transition ellipsoids that arise in Hamiltonian systems
subject to dissipative forces, and which play a crucial role in characterizing
trajectories that evolve across an index-1 saddle point of the underlying
potential energy surface.
