Lagrangian coherent structures: Generalizing stable and unstable manifolds to non-autonomous dynamical systems

In this paper, Lagrangian Coherent Structures are presented as the extensions of stable and un-stable manifolds to non-autonomous dynamical systems. Theoretical and numerical illustrations of Lagrangian Coherent Structures are given, and the results used to analyze some simple sys-tems, both autonomous and non-autonomous. Finally, the application of Lagrangian Coherent Structures to empirical fluid flows is discussed. 1 Dynamical Systems: Background The theory of dynamical systems combines quantitative and qualitative analysis of systems of dif-ferential equations, and was first expounded by Poincare ́ [1] in 1890 during his work on celestial mechanics. Many other dynamical systems also arise as models of real-world behavior, including oscillating pendulums, population dynamics and fluid flows. For ‘simple ’ systems1 there are many analytical concepts such as fixed points, periodic orbits, manifolds and attractors with which to analyze both localized and global features of the system. However, many of these features do not appear in systems with aperiodic time dependence. This paper discusses the use of Lagrangian Coherent Structures in non-autonomous dynamical systems as analogs of stable and unstable man