10 Additional Reading

While it is assumed that the reader has had some exposure to classical dynamical systems concepts, there are a number of good books out there that cover the basic theory. The book by Strogatz (1994) provides an excellent introduction to dynamical systems and chaos and reads very fluently. The text by Verhulst (1996) provides a concise coverage of several main dynamical systems concepts and is also a good introductory book. The very popular book by Guckenheimer and Holmes (1983) is a great reference for dynamical systems theory. Wiggins (1992) and Ottino (1989) are two texts dedicated to studying transport in dynamical systems, and provide a good foundation for understanding this area of dynamical systems.

This tutorial presents a technique for analyzing the global flow geometry of time-dependent dynamical systems. Since a main source of such systems are fluid flow problems, emphasis was placed on such applications, however FTLE's and LCS are also applicable to other systems as well. It has long been recognized that flows with general time dependence admit emergent patterns which influence the transport of tracers. These structures are often referred to generically as coherent structures in the fluid mechanics literature. When coherent structures are studied in terms of quantities derived from fluid trajectories they are often named Lagrangian coherent structures (LCS).  There is a vast body of literature on coherent structures in fluid mechanics that we will not attempt to overview here, however Haller and Yaun (2000) and Provenzale (1999) provide many useful references on this subject and discuss some of the various approaches to understanding transport and are nice papers to read.

As noted by Haller and Yuan (2000), in much of the literature coherent structures are often vaguely defined, making the analysis and algorithmic detection of such structures difficult. The motivation for the series of papers Haller and Poje(2000),  Haller and Yuan (2000), Haller (2001), and Haller (2002) was to give a precise definition of LCS for general time dependent systems defined, perhaps, only over a finite time interval. Haller provides a hyperbolicity time approach which gives criteria for the existence of LCS based on invariants of the gradient of the velocity field evaluated along fluid trajectories. Using this approach, LCS are defined by local extrema of hyperbolicity time fields. Additionally, for the sake of comparison, Haller (2001)  gives an alternative definition of LCS in Sec 2.3 of that paper as local extrema of the finite-time Lyapunov exponent field. He goes on to show the strong correspondence between hyperbolicity time fields and finite-time Lyapunov exponent fields for steady and forced ABC (Arnold-Beltrami-Childress) flows. While this tutorial does not stick to Haller's hyperbolicity time approach, these papers helped motivate a lot of the work that is shown in this tutorial and are recommended.

Doerner et al. (1999) discussed how level contours of finite-time Lyapunov exponent fields can approximate stable manifolds for time-independent and periodic systems. In the realm of general time dependent systems, Pierrehumbert (1991) and  Pierrehumbert and Yang (1993) used finite-time Lyapunov exponent fields to capture chaotic mixing regions and transport barriers from atmospheric data. They used these fields to visualize the existence of structures similar to classical invariant tori that are well-documented in time-periodic flows. In the paper  by von Hardenberg et al. (2000), a similar approach is used for studying atmospheric vortices.

Many of the ideas covered in this tutorial are explained by Shadden, et al (2005). There the approach is slightly more formal and it is not assumed that the dynamical system is smooth. When dealing with discrete data, the data can often be interpolated to provide an adequate level of continuity, however most interpolators become increasingly complex when the level of continuity is increased. Therefore it is not reasonable to assume that quantities derived from the data will be infinitely smooth, so one must take care to make sure the quantities are well behaved.  Application of the ideas presented in this tutorial can be found in Voth et al (2002), Lekien et al (2004), Coulliette, et al (2004), and Inanc et al (2005).  It should be mentioned that there exists other methods for locating hyperbolic structures like LCS using techniques other than FTLE's, such as uniformly hyperbolic trajectories Yuan et al (2002), Rogerson et al (1999), or Poje and Haller (1999) or exponential dichotomies Coppel (1978) or Mancho et al (2003), however, it has been the author's experience that FTLE's are often better suited when working with discrete data sets defined over finite time intervals, however it is interesting to learn about other techniques.