Student(s):
Pavel Zelinsky
Dates of Involvement:
Spring '09- Fall '09
Faculty Advisor(s):
Jim Meiss
Graduate Mentor:
Brock Mosovsky
FTLE of the Standard Map
Background
The Finite Time Lyapunov Exponent (FTLE) field is often used to study fluid flows. The idea behind an FTLE field is that by computing the flow map for a fixed integration time T we can calculate the Jacobian at any point in the flow, which will give us information about how nearby trajectories separate. Then by identifying regions in which nearby particles separate exponentially fast as the flow evolves, we get an approximation of the underlying stable and unstable manifolds.
Manifolds are of interest because they act as separatrices, forming flux barriers between dynamically distinct regions of the flow. Thus, identification of the stable and unstable manifolds can give information about mixing and transport, two areas of study with far-reaching applications, and can help us to better understand the geometry of the flow. While there is a well-developed theory for the identification and computation of stable and unstable manifolds in autonomous (time independent) and time-periodic systems, much of this theory breaks down when applied to cases where the vector field depends aperiodically on time.
For these fully time-dependent systems, the FTLE can be a useful tool in extracting approximate invariant manifolds. By computing the FTLE over the flow domain, we can identify the approximate invariant manifolds as ridges, or generalized local maxima, in the scalar FTLE field. The sharper and more well-defined the ridge, the more fully it inhibits fluid flux across it.
This analysis is mainly used in fluid flow because we are interested in seeing where fluid particles will travel with time, and ridges of high FTLE values can help to uncover structure and geometry in the flow that could not be observed otherwise. The actual calculation I used for the FTLE was taken from a paper by Shawn Shadden, Francois Lekien, and Jerrold Marsden, and it is as follows:
Let 
be the function of the fluid that takes the vector 
at time t0 and advects the particle to time t0 + T. We define the Jacobian as
and the Cauchy-Green deformation tensor as the symmetric matrix
Then the FTLE is defined as
Discrete-Time FTLE
After studying the notion of FTLE described above as it applies to several examples, I became interested in the generalization of this concept to discrete-time systems, or maps.
All of the proofs in Shadden’s paper involve continuous time, but all the computational steps for finding an FTLE value truly only require the flow to be continuous in space and have a continuous derivative. Because of this, I interpreted the “integration time” T as simply the number of iterations n for the discrete-time map. We also note that the standard map is independent of iteration number, that is, the map itself does not depend on the current iteration, but only location within phase space. The validity of using the iteration count n in place of the integration time T can be additionally shown by examining how we compute the FTLE field in the continuous-time case; we do so by a numerical integration method such as Euler or Runge-Kutta which turns the continuous system into a discrete-time system and creates a conversion from time into the number of iterations. With such a definition of time for a map and since maps are continuous in space and have a continuous second derivative, maps can be used in place of a fluid flow.
The particular map I used is the Standard Map. This iteration independent map describes a kicked rotor system and is ideal for inspection with the FTLE field because it is area preserving yet highly chaotic. Additionally there are stable regions within the chaos that have been studied in the past and have been proven to sometimes be fractal. A fluid flow with such properties should produce a wide variety of FTLE values across the field. The definition of the Standard Map I used is:
Another useful property of the Standard Map is that we can find the flow map Jacobian exactly by taking the derivative theoretically and doing a series of chain rules, thus eliminating the need for numerical differentiation. A recurrence relation for the form of the jacobian after n iterations is:
Simulation I have been working on a program that applies the computation of discrete-time FTLE to the Standard Map and produces an interactive simulation. The Standard Map is particularly interesting because it is area-preserving while still being chaotic. This program is useful because it gives an automated way of seeing how the map behaves on a global scale.
About Pavel Zelinsky:
Pavel Zelinsky is a Sophomore majoring in Computer Science and Applied Math with an emphasis on Gaming and Simulation.
