# Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems

## Abstract

**:**

## 1. Introduction

- To establish the relationships between long-time FTNMs and CLVs, and their growth rates, for aperiodic, as well as periodic, chaotic dynamical systems;
- To deduce these properties for systems with degenerate as well as nondegenerate Lyapunov spectra;
- To examine and propose methods for the calculation of CLVs and Lyapunov exponents that allow for degenerate Lyapunov spectra.

## 2. Dynamical Equations

## 3. Eigenvectors and Eigenvalues of the Propagator

#### 3.1. Finite-Time Normal Modes

#### 3.2. Finite-Time Adjoint Modes

## 4. Floquet Vectors

## 5. Singular Vectors

## 6. Lyapunov Vectors and Exponents and the Oseledec Theorem

## 7. Relationships between Covariant Lyapunov Vectors and Orthogonal Lyapunov Vectors

#### 7.1. Construction of Orthonormal Lyapunov Vectors from Covariant Lyapunov Vectors

#### 7.2. Construction of Covariant Lyapunov Vectors from Orthonormal Lyapunov Vectors

#### 7.3. Degenerate Orthonormal Lyapunov Vectors

## 8. Dynamics in FTNM-Space

#### Tangent Linear Equation and Propagator in FTNM-Space

## 9. Covariant Lyapunov Vectors for Periodic Systems

#### 9.1. Dynamics of SVs and OLVs in FTNM-Space

#### 9.2. Construction of CLVs from OLVs in FTNM-Space

#### 9.3. Construction of OLVs from CLVs in FTNM-Space

## 10. Covariant Lyapunov Vectors for Aperiodic Systems

#### 10.1. Dynamics in FTNM-Space

#### 10.2. Transformation to the Original Phase-Space

**Φ**

^{±1}, which depend on the reference trajectory and time, is sufficient for the convergence of the Lyapunov exponents (Appendix D, Equations (A73) to (A75)). The proof of the Oseledec Theorem 4 [72], which establishes the Lyapunov exponents and Oseledec subspaces, also uses a diagonalization of the propagator cocycle for dynamics in the subspaces. Oseledec used a homologous transformation of the propagator cocycle with the transformation matrices

**Φ**

^{±1}satisfying the Lyapunov condition (Appendix C, Equations (A54) and (A55)). The Lyapunov condition in Equation (A55), which means that

**Φ**

^{±1}have no asymptotic exponential growth, is slightly less stringent than the boundedness constraint in Equation (A73). With either condition, the relationship in Equation (113), between

**ψ**

^{n}(t) and

**ϕ**

^{n}(t) becomes equality for τ

_{±}→±∞. As noted, Δ

_{±}= γ

_{±}τ

_{±}and 1 > γ

_{±}>0, so that Δ

_{±}→±∞ and we also have ${\mathsf{\Lambda}}_{r}^{n}({\tau}_{+},{\tau}_{-})={{\displaystyle \sum _{\_}}}^{n}({\tau}_{+},{\tau}_{-})\to {\mathcal{L}}^{n}$.

## 11. Calculation of Dynamical Vectors and Metric Entropy Production

#### 11.1. Lyapunov Vectors

#### 11.2. Degeneracy and Nondegeneracy

#### 11.3. Finite-Time Normal Modes and Arnoldi Methods

#### 11.4. Metric Entropy Production

## 12. Discussion and Conclusions

- 1.
- The covariant properties of FTNMs, ${\mathit{\varphi}}^{n}(t)$, in the time interval ${t}_{f}\ge t\ge {t}_{0}$ are known and determined by ${\mathit{\varphi}}^{n}(t)=\mathbf{G}(t,{t}_{0}){\mathit{\varphi}}^{n}({t}_{0})$, as in Equation (19). We show that they also satisfy the eigenvalue problem $\mathbf{G}(t+T,{t}_{0}+T)\mathbf{G}({t}_{0}+T,t){\mathit{\varphi}}^{n}(t)={\lambda}^{n}{\mathit{\varphi}}^{n}(t)$ in Equation (21) where $\mathbf{G}(t+T,{t}_{0}+T)\equiv \mathbf{G}(t,{t}_{0})$ by definition as noted in Equation (22). The propagator is in general discontinuous at ${t}_{0}+T={t}_{f}$. In the case where it is continuous the FTNMs become Floquet vectors as noted in Equation (37). Indeed, in the efficient Arnoldi algorithm of Wei and Frederiksen [27], discussed in Section 11, the leading FTNMs are constructed by recycling perturbations from ${t}_{f}={t}_{0}+T$ to ${t}_{0}$.
- 2.
- In FTNM-space, the right, or initial, SVs ${\underset{\_}{v}}^{n}({t}_{0})$ and left, or final, SVs ${\underset{\_}{u}}^{n}({t}_{f})$, for $n=1,2,\dots ,N$, are, like the FTNMs, ${\underset{\_}{\mathit{\varphi}}}^{n}({t}_{0})$ and ${\underset{\_}{\mathit{\varphi}}}^{n}({t}_{f})$, proportional to the unit vectors ${e}^{n}$ as shown in Equations (73) and (74). This is because the propagator between ${t}_{0}$ and ${t}_{f}$ in FTNM-space is normal, in fact diagonal (Equation (65)). A particular consequence is that in FTNM-space the singular value exponents are equal to the real part of the FTNM exponents: ${{\displaystyle \sum _{\_}}}^{n}({t}_{f},{t}_{0})={\mathsf{\Lambda}}_{r}^{n}({t}_{f},{t}_{0})$ as noted in Equation (76).
- 3.
- The relationships, based on the Oseledec theorem [72], for the construction of OLVs from CLVs (Equations (53) and (54)) and importantly the determination of CLVs from OLVs (Equations (58) and (59)) and their approximations by long-time SVs in Section 7 greatly simplify at critical times in FTNM-space as shown in Section 9 and Section 10. This is because of the diagonalization of the propagator in result 2 above.
- 4.
- In Appendix C, the propagator $\mathbf{G}({t}_{f},{t}_{0})$ in the original $x$-space and the propagator $\underset{\_}{\mathbf{G}}({t}_{f},{t}_{0})$ in FTNM-space have both been shown to be homologous to the stretching propagator $\tilde{\underset{\_}{\mathbf{G}}}({t}_{f},{t}_{0})$ with the transformation matrices subject to the Lyapunov condition in Equation (A55). This ensures that if the results of Oseledec’s [72] four theorems, including his multiplicative ergodic theorem 4, apply for a propagator in any of the phase-spaces then they hold for all these cohomologous propagator cocycles.
- 5.
- For periodic systems, the results 3 and 4 above have been used in Section 9 to deduce the links of FTNMs to OLVs and CLVs, from Equations (53) to (59), and the equivalence of Floquet vectors and CLVs. Moreover, the Lyapunov exponents ${\mathcal{L}}^{n}={\mathsf{\Lambda}}_{r}^{n}({t}_{f},{t}_{0})={{\displaystyle \sum _{\_}}}^{n}({t}_{f},{t}_{0})={{\displaystyle \sum _{\_}}}^{n}({t}_{f},-\infty )={{\displaystyle \sum _{\_}}}^{n}({t}_{0},-\infty )={{\displaystyle \sum _{\_}}}^{n}(\infty ,{t}_{0})={{\displaystyle \sum _{\_}}}^{n}(\infty ,{t}_{f})$, as given in Equation (92). This applies for systems with both real and complex Floquet exponents and nondegenerate and degenerate Lyapunov spectra.
- 6.
- In the case of aperiodic systems, including with degenerate Lyapunov spectra, the results 3 and 4 above have been used in Section 10 to show that in the interval ${\tau}_{+}>{\Delta}_{+}\ge t\ge {\Delta}_{-}>{\tau}_{-}$, CLVs are closely approximated by FTNMs, ${\mathit{\psi}}^{n}(t)\approx {\mathit{\varphi}}^{n}(t)$, for large $\left|{\tau}_{\pm}\right|$ with equality as ${\tau}_{\pm}\to \pm \infty $. Moreover, in FTNM-space the singular value exponent and FTNM growth rates are equal and approximate the Lyapunov exponent ${{\displaystyle \sum _{\_}}}^{n}({\tau}_{+},{\tau}_{-})={\mathsf{\Lambda}}_{r}^{n}({\tau}_{+},{\tau}_{-})\doteq {\mathcal{L}}^{n}$ with equality as ${\tau}_{\pm}\to \pm \infty $.
- 7.
- An alternative way of establishing the results in 5 and 6 is presented in Appendix E where FTNMs are orthogonalized using the Gram-Smidt method and the long-time limits of the orthonormal vectors are considered particularly in FTNM phase-space.
- 8.
- Finite-time generalizations of the Kolmogorov-Sinai entropy production [8,9] and the Kaplan-Yorke conjectured Hausdorff dimension [97] have been proposed based on the FTNM exponents. The expressions, like the FTNMs, are norm-independent and may be reasonably accurate for ensembles of perturbations as noted in Section 11 although it is possible that intramodal and intermodal interference effects could contribute to individual perturbation growth [79,80].

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Ordering of Eigenvalues and Eigenvectors

## Appendix B. The Oseledec Multiplicative Ergodic Theorem

#### Appendix B.1. Lyapunov Exponents

#### Appendix B.2. Oseledec Operators

#### Appendix B.3. Oseledec Subspaces—Nondegenerate Case

#### Appendix B.4. Oseledec Subspaces—Degenerate Case

#### Appendix B.5. Oseledec Operators in FTNM-Space

## Appendix C. Lyapunov Homologous Propagator Cocycles

## Appendix D. Phase-Space Transformation of CLVs and Lyapunov Exponents

## Appendix E. Orthogonalization of FTNMs and Asymptotic Behaviour

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**MDPI and ACS Style**

Frederiksen, J.S. Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems. *Entropy* **2023**, *25*, 244.
https://doi.org/10.3390/e25020244

**AMA Style**

Frederiksen JS. Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems. *Entropy*. 2023; 25(2):244.
https://doi.org/10.3390/e25020244

**Chicago/Turabian Style**

Frederiksen, Jorgen S. 2023. "Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems" *Entropy* 25, no. 2: 244.
https://doi.org/10.3390/e25020244