[Paper Review] Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and vectors

Kenneth D. Mease, Ufuk Topcu+2 more|arXiv (Cornell University)|Jul 1, 2008

Quantum chaos and dynamical systems3 citations

TL;DR

This paper proposes a finite-time Lyapunov analysis (FTLA) framework to diagnose two-timescale nonlinear dynamics and compute center manifold structures in finite-dimensional autonomous systems without requiring equilibrium points or normal forms. By using finite-time Lyapunov exponents and vectors to define a slow-fast splitting of the tangent bundle and enforcing invariance-based orthogonality conditions, the method locates normally hyperbolic center manifolds with high accuracy, validated through multiple numerical examples including a nonlinear oscillator and a flight dynamics model.

ABSTRACT

Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for example, finite-time Lyapunov analysis relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. The splitting is used to locate points on normally hyperbolic center manifolds. Determining manifolds from tangent bundle structure is more generally applicable than approaches, such as the singular perturbation method, that require special normal forms or other a priori knowledge. The use, features, and accuracy of the approach are illustrated via several detailed examples.

Motivation & Objective

• To develop a method for diagnosing two-timescale behavior in nonlinear dynamical systems without relying on equilibrium points or singular perturbation normal forms.
• To characterize the tangent bundle structure via finite-time Lyapunov exponents and vectors for identifying slow and fast subspaces.
• To compute normally hyperbolic center manifolds using invariance-based orthogonality conditions derived from finite-time dynamics.
• To enable application of manifold-based reduction techniques to finite-time, non-equilibrium system behavior in practical engineering contexts such as flight control.

Proposed method

• Define a slow-fast splitting of the tangent bundle using finite-time Lyapunov exponents (FTLEs) and vectors (FTLVs), with convergence analysis for the FTLVs.
• Use the FTLVs to identify the slow (center) and fast (stable/unstable) subspaces in state space regions, even away from equilibria.
• Apply invariance-based orthogonality conditions to locate points on the center manifold by solving for vectors orthogonal to fast dynamics in the tangent space.
• Implement an iterative algorithm with forward and backward averaging times that are adaptively increased to improve convergence of FTLV estimates.
• Use stopping criteria based on relative changes in state variables and angles between vector projections to ensure numerical accuracy.
• Automate the averaging time selection by iteratively increasing forward and backward integration times in outer-loop iterations.

Experimental results

Research questions

• RQ1 Can finite-time Lyapunov exponents and vectors reliably detect two-timescale behavior in nonlinear systems without requiring equilibrium or periodic orbit assumptions?
• RQ2 How can the tangent bundle structure be decomposed into slow and fast subspaces using finite-time dynamics, and what convergence properties do the associated FTLVs exhibit?
• RQ3 To what extent can invariance-based orthogonality conditions accurately locate points on normally hyperbolic center manifolds using only finite-time data?
• RQ4 How does the adaptive adjustment of forward and backward averaging times affect the accuracy and convergence of the manifold approximation?

Key findings

• The finite-time Lyapunov vectors converge exponentially fast, enabling robust identification of slow and fast subspaces even for short integration times.
• The method successfully locates points on the center manifold with a relative error tolerance of 10−6 in the outer-loop convergence criterion.
• For the tested examples, the iterative algorithm required approximately 5 inner iterations per outer loop, with forward and backward averaging times increasing to about 5.0 and 2.0, respectively.
• The computed manifold approximations were consistent across different initial guesses, indicating robustness and convergence to the same invariant manifold.
• The theoretical bound on the convergence rate of the FTLVs was conservative but confirmed exponential convergence, supporting the method's stability.
• The approach outperforms traditional methods by not requiring prior knowledge of normal forms or analytical continuation, enabling application to general nonlinear systems.