# Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Method

## 3. Numerical Simulations

#### 3.1. Methodology

#### 3.1.1. Method 1 (M1)

#### 3.1.2. Method 2 (M2)

#### 3.1.3. Method 3 (M3)

**z**normalized in the case of excessively high or low values of perturbation length $\mathit{z}\left(t\right)$:

#### 3.1.4. Method 4 (M4)

#### 3.1.5. Method 5 (M5)

#### 3.2. Results of Numerical Simulations

#### 3.3. Duffing Oscillator

#### 3.4. The Van der Pol Oscillator

## 4. Largest Lyapunov Exponent (LLE) from Maps

**z**and then next perturbation

**z**from the next points of crossing trajectories through the hyperplane $\pi $. After projection of the difference of the vectors ${\mathit{z}}_{1}-\mathit{z}$ on to the direction of perturbation $\mathit{z}$, one obtains a differential $d\mathit{z}$. It allows for substituting the lengths $z$ and $dz$ into Equation (4) to find $\lambda $ value. Alternatively, $dz$ can be calculated from the difference of the norms of vectors $\left|{\mathit{z}}_{1}\right|-\left|\mathit{z}\right|$. However, in this case, the estimation error is expected to be higher. During the evolution of the system, obtained values have to be averaged and then recalculated according to the error correction analysis presented below.

_{1}#### Error Correction Analysis

**π**, there were calculated $i$ steps of numerical integration. During numerical calculation of LLE, in each integration step, values ${\lambda}_{i}$ are obtained, and then averaged in time in order to obtain LLE. Following reasoning that justified scalar notation of Equation (3), we can continue in the same vein in the case of maps. Then, the value of the proposed indicator for a map is:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Diagram of the largest Lyapunov exponent of the Duffing system and computation times ${t}_{i}$[s] graphs. $\alpha $ = 1, $\beta $ = 0.05, $\omega $ = 0.47.

**Figure 3.**Diagram of efficiencies ${\eta}_{1},{\eta}_{2},{\eta}_{4},{\eta}_{5}$, of LLE computations of the Duffing system. $\alpha $ = 1, $\beta $ = 0.05, $\omega $ = 0.47.

**Figure 4.**Diagram of accuracy of LLE computations, of the Duffing system. $\alpha $ = 1, $\beta $ = 0.05, $\omega $ = 0.47.

**Figure 5.**Diagram of the largest Lyapunov exponent of the Van der Pol system and computation times ${t}_{i}$[s] graphs. $q$ = 12.95, $\omega $ = 4.64.

**Figure 6.**Diagram of efficiencies ${\eta}_{1},{\eta}_{2},{\eta}_{4},{\eta}_{5}$ of LLE computations of the Van der Pol system. $q$ = 12.95, $\omega $ = 4.64.

**Figure 7.**Diagram of the accuracy of LLE computations of the Van der Pol system. $q$ = 12.95, $\omega $ = 4.64.

**Figure 10.**Time series of the largest Lyapunov exponent of the Duffing system. $\alpha $ = 1, $\beta $ = 0.05, $\omega $ = 0.47.

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**MDPI and ACS Style**

Dabrowski, A.; Sagan, T.; Denysenko, V.; Balcerzak, M.; Zarychta, S.; Stefanski, A.
Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method. *Materials* **2021**, *14*, 7197.
https://doi.org/10.3390/ma14237197

**AMA Style**

Dabrowski A, Sagan T, Denysenko V, Balcerzak M, Zarychta S, Stefanski A.
Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method. *Materials*. 2021; 14(23):7197.
https://doi.org/10.3390/ma14237197

**Chicago/Turabian Style**

Dabrowski, Artur, Tomasz Sagan, Volodymyr Denysenko, Marek Balcerzak, Sandra Zarychta, and Andrzej Stefanski.
2021. "Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method" *Materials* 14, no. 23: 7197.
https://doi.org/10.3390/ma14237197