Lagrangian Coherent Structures
Lagrangian Coherent Sturctures are defined as state space boundaries in the finite time Lyapunov exponent (FTLE) field. These can be used to locate the boundaries that separate regions of state space in which qualatatively different behavior is exhibited. The approach uses similar methods to calcuate the FTLE to those used in the stability paragraph above. However, unlike previous approaches that average the FTLE over the entier time series, in this method an FTLE field is generated based on the initial location of each pair of trajectories in state space. This results an entire field of data rather than a single scalar value for the system.
Deterministic Inverted Pendulum:
The state transition matrix method is used to calculate the FTLE field from a regular grid of points centered around the upright vertical position. In the FTLE field is shown below, an LCS (red ridge) can be seen to separate the stable from the unstable regions of state space.
Matlab Code:
Code is freely available for download: Inverted Pendulum SimulationReference:
Tanaka ML, Ross SD, “Separatrices and Basins of Stability from Time Series Data: An Application to Biodynamics”, Nonlinear Dynamics, in presshttp://dx.doi.org/10.1007/s11071-008-9457-9