Math. Model. Nat. Phenom.
Volume 10, Number 3, 2015Model Reduction
|91 - 104
|22 June 2015
Flow Structure Identification for Nonlinear Dynamical Systems via Finite-Time Lyapunov Analysis
Department of Mechanical and Aerospace Engineering, University of California
Irvine CA, 92697, USA
Corresponding author. E-mail: email@example.com
Identifying and characterizing geometric structure in the flow of a nonlinear dynamical system can facilitate understanding, model simplification, and solution approximation. The approach addressed in this paper uses information from finite-time Lyapunov exponents and vectors associated with the tangent linear dynamics. We refer to this approach as finite-time Lyapunov analysis (FTLA). FTLA identifies the potential for flow structure based on the stability and timescales implied by the spectrum of finite-time Lyapunov exponents. The corresponding Lyapunov vectors provide a basis for representing a splitting of the tangent space at phase points consistent with the splitting of the spectrum. A key property that makes FTLA viable is the exponential convergence of the splitting as the finite time increases. Tangency conditions for the vector field are used to determine points on manifolds of interest. The benefits of the FTLA approach are the dynamical model need not be in a special normal form, the manifolds of interest need not be attracting nor of known dimension, and the manifolds need not be associated with a fixed point or periodic orbit.
After a brief review of FTLA, it is applied to spacecraft stationkeeping around a libration point in the circular restricted three-body problem. This application requires locating the stable and unstable subspaces at points on periodic and aperiodic orbits. For the periodic case, the FTLA subspaces are shown to agree with the Floquet subspaces; for the quasi-periodic case, the accuracy of the FTLA subspaces is demonstrated by simulation.
Mathematics Subject Classification: 34D08 / 34D35 / 34N05 / 37D05 / 37D25 / 37D30 / 37J15 / 37M25
Key words: finite-time Lyapunov exponents / finite-time Lyapunov vectors / flow structure / nonlinear dynamical systems / timescales / circular restricted three-body problem / spacecraft stationkeeping
© EDP Sciences, 2015
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.