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

^{1}

^{3}S

^{2}, The Clarkson Center for Complex Systems Science, Clarkson University, Potsdam, NY 13699, USA

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^{*}

## Abstract

**:**

## 1. Significance

## 2. Instantaneous Lyapunov Exponent Analysis

#### 2.1. Instantaneous Attraction and Repulsion Rates

#### 2.2. Finite-Time Lyapunov Exponent

## 3. On Evolution of Observations, the Koopman Operator and Its Eigenfunction PDE

## 4. One-Dimensional Vector Fields

## 5. Two-Dimensional Non-Linear Saddle Flow

## 6. General Two-Dimensional Vector Fields

## 7. Polynomial Vector Fields

#### 7.1. Quadratic Vector Fields

#### 7.2. Cubic Vector Fields

#### 7.3. Cubic Vector Field Example

#### 7.4. Cubic Vector Field Transformation to Simplify

#### 7.5. Properties of KEIGs of 2-Dimensional Cubic Vector Fields

#### 7.6. A Family of Polynomial Vector Fields with a KEIG Attraction Rate

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Verification of the Form of the Koopman Eigenfunctions

## Appendix B. Arbitrary Functions g(x) Can Be Written as an Infinite Series of Koopman Eigenfunctions

## References

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**Figure 1.**Schematic of the effect of the co-dimension 1 manifold, corresponding to the trench of the attraction rate, ${s}_{1}$, at an initial time, on a small parcel of phase space. The image of the trench under the time-t flow map, ${\mathbf{F}}_{t}$, is also shown.

**Figure 2.**The action of a Koopman operator (composition operator) is to extract the value of a measurement function downstream, (

**Left**) Equation (13), but (

**Right**) for eigenvalue and eigenfunction pair (KEIGS) $(\lambda ,{\varphi}_{\lambda}\left(\mathbf{x}\right))$ that function has the property Equation (14) effectively interpreting the change of ${\varphi}_{\lambda}$along an orbit as if the dynamics are linear even if the flow may be non-linear in its phase space M.

**Figure 3.**A general eigenfunction, that is, Equation (19) is a solution of Equation (18) is defined in terms of measuring the initial data $h\left(s\right)$ on the data surface $\Lambda $, and then for each $\mathbf{x}$ there is a unique point where that measurement is taken. Then, the solution ${\varphi}_{\lambda}$ is then a linear scaling of that measurement, by the time of the pull back.

**Figure 4.**Phase portrait for vector field (80) in $({x}_{1},{x}_{2})$ coordinates for parameters $(\lambda ,c)=(-1,-1)$. The attraction rate field for this vector field is also a Koopman eigenfunction (KEIG). The ${x}_{2}$-axis $\{{x}_{1}=0\}$, as trench of the attraction rate field, is an attracting instantaneous Lyapunov exponent structure (iLES).

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Bollt, E.M.; Ross, S.D. Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction? *Mathematics* **2021**, *9*, 2731.
https://doi.org/10.3390/math9212731

**AMA Style**

Bollt EM, Ross SD. Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction? *Mathematics*. 2021; 9(21):2731.
https://doi.org/10.3390/math9212731

**Chicago/Turabian Style**

Bollt, Erik M., and Shane D. Ross. 2021. "Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?" *Mathematics* 9, no. 21: 2731.
https://doi.org/10.3390/math9212731