# Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?

@article{Bollt2021IsTF, title={Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?}, author={Erik M. Bollt and Shane D. Ross}, journal={Mathematics}, year={2021} }

This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis, and the global operator theoretic Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link…

## One Citation

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## References

SHOWING 1-10 OF 41 REFERENCES

### Global Stability Analysis Using the Eigenfunctions of the Koopman Operator

- MathematicsIEEE Transactions on Automatic Control
- 2016

The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles.

### Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Geometry

- MathematicsJournal of Nonlinear Science
- 2019

A spectral expansion for general linear autonomous dynamical systems with analytic observables, and the notion of generalized eigenfunctions of the associated Koopman operator is defined, and isostables for a general class of nonlinear systems are defined.

### Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows

- Mathematics
- 2005

### Finite-time Lyapunov exponents in the instantaneous limit and material transport

- GeologyNonlinear Dynamics
- 2020

Lagrangian techniques, such as the finite-time Lyapunov exponent (FTLE) and hyperbolic Lagrangian coherent structures, have become popular tools for analyzing unsteady fluid flows. These techniques…

### Bifurcations and Periodic Orbits of Vector Fields

- Mathematics
- 1993

Preface. Complex Foliations arising from Polynomial Differential Equations C. Camacho. Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular…

### The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds.

- MathematicsChaos
- 2010

The FTLE computational method is modified to accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.

### Analysis of Fluid Flows via Spectral Properties of the Koopman Operator

- Physics
- 2013

This article reviews theory and applications of Koopman modes in fluid mechanics. Koopman mode decomposition is based on the surprising fact, discovered in Mezic (2005), that normal modes of linear…

### A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

- MathematicsJ. Nonlinear Sci.
- 2015

This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation.

### Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

- Mathematics
- 2005

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of…