# 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

## References

- Hénon, M.; Heiles, C. The applicability of the third integral of the motion: Some numerical results. Astron. J.
**1964**, 69, 73–79. [Google Scholar] [CrossRef] [Green Version] - Liao, H. Novel gradient calculation method for the largest Lyapunov exponent of chaotic systems. Nonlinear Dyn.
**2016**, 85, 1377–1392. [Google Scholar] [CrossRef] - Peixoto, M.L.; Nepomuceno, E.G.; Martins, S.A.; Lacerda, M.J. Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm. Chaos Solitons Fractals
**2018**, 112, 36–43. [Google Scholar] [CrossRef] - Zhou, S.; Wang, X. Identifying the linear region based on machine learning to calculate the largest Lyapunov exponent from chaotic time series. Chaos Interdiscip. J. Nonlinear Sci.
**2018**, 28, 123118. [Google Scholar] [CrossRef] - Zhou, S.; Wang, X.; Wang, Z.; Zhang, C. A novel method based on the pseudo-orbits to calculate the largest Lyapunov exponent from chaotic equations. Chaos Interdiscip. J. Nonlinear Sci.
**2019**, 29, 033125. [Google Scholar] [CrossRef] [PubMed] - Shuang, Z.; Yong, F.; Wen-Yuan, W.; Wei-Hua, W. A novel method based on the fuzzy C-means clustering to calculate the maximal Lyapunov exponent from small data. Acta Phys. Sin.
**2016**, 65, 020502. [Google Scholar] - Liao, H.; Wu, W. The Reduced Space Shooting Method for Calculating the Peak Periodic Solutions of Nonlinear Systems. J. Comput. Nonlinear Dyn.
**2018**, 13, 061001. [Google Scholar] [CrossRef] - Zhang, Y.; Chen, D.; Guo, D.; Liao, B.; Wang, Y. On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn.
**2014**, 79, 983–1003. [Google Scholar] [CrossRef] - Balcerzak, M.; Dabrowski, A.; Blazejczyk–Okolewska, B.; Stefanski, A. Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach. Mech. Syst. Signal Process.
**2020**, 141, 106734. [Google Scholar] [CrossRef] - Liao, H. Stability Analysis of Duffing Oscillator with Time Delayed and/or Fractional Derivatives. Mech. Based Des. Struct. Mach.
**2015**, 44, 283–305. [Google Scholar] [CrossRef] - Śmiechowicz, W.; Loup, T.; Olejnik, P. Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation. Math. Comput. Appl.
**2019**, 24, 90. [Google Scholar] [CrossRef] [Green Version] - Liao, H. Nonlinear Dynamics of Duffing Oscillator with Time Delayed Term. Comput. Modeling Eng. Sci.
**2014**, 103, 155–187. [Google Scholar] - Stefanski, A.; Pikunov, D.; Balcerzak, M.; Dabrowski, A. Synchronized chaotic swinging of parametrically driven pendulums. Int. J. Mech. Sci.
**2020**, 173, 105454. [Google Scholar] [CrossRef] - Kengne, J.; Njitacke Tabekoueng, Z.; Kamdoum Tamba, V.; Nguomkam Negou, A. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos Interdiscip. J. Nonlinear Sci.
**2015**, 25, 103126. [Google Scholar] [CrossRef] - Wadduwage, D.P.; Wu, C.Q.; Annakkage, U.D. Power system transient stability analysis via the concept of Lyapunov Exponents. Electr. Power Syst. Res.
**2013**, 104, 183–192. [Google Scholar] [CrossRef] - Rajagopal, K.; Jafari, S.; Karthikeyan, A.; Srinivasan, A.; Ayele, B. Hyperchaotic Memcapacitor Oscillator with Infinite Equilibria and Coexisting Attractors. Circuits Syst. Signal Process.
**2018**, 37, 3702–3724. [Google Scholar] [CrossRef] - Dingwell, J.; Cusumano, J.P. Nonlinear time series analysis of normal and pathological human walking. Chaos Interdiscip. J. Nonlinear Sci.
**2000**, 10, 848–863. [Google Scholar] [CrossRef] - Eteme, A.S.; Tabi, C.B.; Ateba, J.F.B.; Ekobena, H.P.F.; Mohamadou, A.; Kofane, T.C.; Fouda, H.P.E. Neuronal firing and DNA dynamics in a neural network. J. Phys. Commun.
**2018**, 2, 125004. [Google Scholar] [CrossRef] - Cao, J.; Lu, J. Adaptive synchronization of neural networks with or without time-varying delay. Chaos Interdiscip. J. Nonlinear Sci.
**2006**, 16, 013133. [Google Scholar] [CrossRef] - Zhou, S.; Feng, Y.; Wu, W.-Y.; Li, Y.; Liu, J. Low-dimensional chaos and fractal properties of long-term sunspot activity. Res. Astron. Astrophys.
**2013**, 14, 104–112. [Google Scholar] [CrossRef] - Shuang, Z.; Yong, F.; Wen-Yuan, W. Chaos and fractal properties of solar activity phenomena at the high and low latitudes. Acta Phys. Sin.
**2015**, 64, 249601. [Google Scholar] - Iwaniec, J.; Uhl, T.; Staszewski, W.J.; Klepka, A. Detection of changes in cracked aluminium plate determinism by recurrence analysis. Nonlinear Dyn.
**2012**, 70, 125–140. [Google Scholar] [CrossRef] - Panahi, S.; Jafari, S.; Khalaf, A.J.M.; Rajagopal, K.; Pham, V.; Alsaadi, F.E. Complete dynamical analysis of a neuron under magnetic flow effect. Chin. J. Phys.
**2018**, 56, 2254–2264. [Google Scholar] [CrossRef] - Etémé, A.S.; Tabi, C.B.; Mohamadou, A.; Kofané, T.C. Long-range memory effects in a magnetized Hindmarsh-Rose neural network. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 84, 105208. [Google Scholar] [CrossRef] - Etémé, A.S.; Tabi, C.B.; Mohamadou, A. Synchronized nonlinear patterns in electrically coupled Hindmarsh–Rose neural networks with long-range diffusive interactions. Chaos Solitons Fractals
**2017**, 104, 813–826. [Google Scholar] [CrossRef] - Ornelas-Tellez, F.; Sanchez, E.N.; Loukianov, A.G.; Navarro-Lopez, E.M. Speed-gradient inverse optimal control for discrete-time nonlinear systems. In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA, 12–15 December 2011; IEEE: New York, NY, USA, 2011; pp. 290–295. [Google Scholar] [CrossRef]
- Hegger, R.; Kantz, H.; Schreiber, T. Practical implementation of nonlinear time series methods: The TISEAN package. Chaos Interdiscip. J. Nonlinear Sci.
**1999**, 9, 413–435. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Balcerzak, M.; Sagan, T.; Dabrowski, A.; Stefanski, A. Fast and simple Lyapunov Exponents estimation in discontinuous systems. Eur. Phys. J. Spec. Top.
**2020**, 229, 2167–2181. [Google Scholar] [CrossRef] - Pikunov, D.; Stefanski, A. Numerical analysis of the friction-induced oscillator of Duffing’s type with modified LuGre friction model. J. Sound Vib.
**2019**, 440, 23–33. [Google Scholar] [CrossRef] - Balcerzak, M.; Dabrowski, A.; Stefański, A.; Wojewoda, J. Spectrum of Lyapunov exponents in non-smooth systems evaluated using orthogonal perturbation vectors. In Proceedings of the MATEC Web of Conferences, International Conference on Engineering Vibration, Sofia, Bulgaria, 4–7 September 2017; EDP Sciences: Les Ulis, France, 2018; Volume 148, p. 10003. [Google Scholar]
- Fuhg, J.N.; Fau, A. Surrogate model approach for investigating the stability of a friction-induced oscillator of Duffing’s type. Nonlinear Dyn.
**2019**, 98, 1709–1729. [Google Scholar] [CrossRef] [Green Version] - Prakash, P.; Rajagopal, K.; Singh, J.P.; Roy, B.K. Megastability, Multistability in a Periodically Forced Conservative and Dissipative System with Signum Nonlinearity. Int. J. Bifurc. Chaos
**2018**, 28, 1830030. [Google Scholar] [CrossRef] - Rajagopal, K.; Duraisamy, P.; Weldegiorgis, R.; Karthikeyan, A. Multistability in Horizontal Platform System with and without Time Delays. Shock. Vib.
**2018**, 2018, 1092812. [Google Scholar] [CrossRef] [Green Version] - Rajagopal, K.; Pham, V.-T.; Tahir, F.R.; Akgül, A.; Abdolmohammadi, H.R.; Jafari, S. A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation. Pramana
**2018**, 90, 52. [Google Scholar] [CrossRef] - Rajagopal, K.; Jafari, S.; Laarem, G. Time-delayed chameleon: Analysis, synchronization and FPGA implementation. Pramana J. Phys.
**2017**, 89, 92. [Google Scholar] [CrossRef] - Rajagopal, K.; Jafari, S.; Akgul, A.; Karthikeyan, A.; Çiçek, S.; Shekofteh, Y. A Simple Snap Oscillator with Coexisting Attractors, Its Time-Delayed Form, Physical Realization, and Communication Designs. Zeitschrift für Naturforschung A
**2018**, 73, 385–398. [Google Scholar] [CrossRef] - Stefański, A.; Kapitaniak, T.; Dąbrowski, A. The largest Lyapunov exponent of dynamical systems with time delay. In Proceedings of the IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, Rome, Italy, 8–13 June 2003; Springer: Dordrecht, The Netherlands, 2005; pp. 493–500. [Google Scholar]
- Ren, S.; Panahi, S.; Rajagopal, K.; Akgul, A.; Pham, V.-T.; Jafari, S. A New Chaotic Flow with Hidden Attractor: The First Hyperjerk System with No Equilibrium. Zeitschrift für Naturforschung A
**2018**, 73, 239–249. [Google Scholar] [CrossRef] - Rajagopal, K.; Jafari, S.; Akgul, A.; Karthikeyan, A. Modified jerk system with self-exciting and hidden flows and the effect of time delays on existence of multi-stability. Nonlinear Dyn.
**2018**, 93, 1087–1108. [Google Scholar] [CrossRef] - Liao, H. Optimization analysis of Duffing oscillator with fractional derivatives. Nonlinear Dyn.
**2014**, 79, 1311–1328. [Google Scholar] [CrossRef] - Huang, S.; Wang, B. Stability and stabilization of a class of fractional-order nonlinear systems for Stability and stabilization of a class of fractional-order nonlinear systems for 0 < α < 2. Nonlinear Dyn.
**2017**, 88, 973–984. [Google Scholar] [CrossRef] - Gomes, C.; Karalis, P.; Navarro-López, E.M.; Vangheluwe, H. Approximated Stability Analysis of Bi-modal Hybrid Co-simulation Scenarios. In Software Engineering and Formal Methods; Lecture Notes in Computer Science; Cerone, A., Roveri, M., Eds.; Springer: London, UK, 2018; Volume 10729, pp. 345–360. [Google Scholar]
- Navarro-López, E.M.; Laila, D.S. Group and Total Dissipativity and Stability of Multi-Equilibria Hybrid Automata. IEEE Trans. Autom.
**2013**, 58, 3196–3202. [Google Scholar] [CrossRef] - Pecora, L.M.; Carroll, T.; Johnson, G.A.; Mar, D.J.; Heagy, J.F. Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos Interdiscip. J. Nonlinear Sci.
**1997**, 7, 520–543. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rajagopal, K.; Naseradinmousavi, P.; Khalaf, A.J.M.; Jafari, S.; Karthikeyan, A.; Çiçek, S. A novel parametrically controlled multi-scroll chaotic attractor along with electronic circuit design. Eur. Phys. J. Plus
**2018**, 133, 354. [Google Scholar] [CrossRef] - Juan, S.; Xiao-Xia, L.; Jin-Hao, Z.; Yu-Zhuo, S.; Yan-Yu, L. Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling. Acta Phys. Sin.
**2017**, 66, 188901. [Google Scholar] - La Guardia, G.G.; Miranda, P.J. Lyapunov exponent for Lipschitz maps. Nonlinear Dyn.
**2018**, 92, 1217–1224. [Google Scholar] [CrossRef] [Green Version] - Balcerzak, M.; Chudzik, A.; Stefanski, A. Properties of generalized synchronization in smooth and non-smooth identical oscillators. Eur. Phys. J. Spec. Top.
**2020**, 229, 2151–2165. [Google Scholar] [CrossRef] - Dabrowski, A.; Glogowski, M.; Kubiak, P. Improving the efficiency of four-stroke engine with use of the pneumatic energy accumulator-simulations and examination. Int. J. Automot. Technol.
**2016**, 17, 581–590. [Google Scholar] [CrossRef] - Etémé, A.S.; Tabi, C.B.; Mohamadou, A. Firing and synchronization modes in neural network under magnetic stimulation. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 72, 432–440. [Google Scholar] [CrossRef] - Dabrowski, A.; Balcerzak, M.; Pikunov, D. Applications of the New Method of the Lyapunov Exponents Estimation. Proceedings of 23rd International Conference on Engineering Mechanics, Svratka, Czech Republic, 15–18 May 2017; pp. 266–269. [Google Scholar]
- Balcerzak, M.; Dąbrowski, A.; Pikunov, D. Tuning the control system of a nonlinear inverted pendulum by means of the new method of Lyapunov exponents estimation. AIP Conf. Proc.
**2018**, 1922, 100016. [Google Scholar] [CrossRef] - Dąbrowski, A.; Jach, A.; Kapitaniak, T. Application of artificial neural networks in parametrical investigations of the energy flow and synchronization. J. Theor. Appl. Mech.
**2010**, 48, 871–896. [Google Scholar] - Dabrowski, A. Energy–vector method in mechanical oscillations. Chaos Solitons Fractals
**2009**, 39, 1684–1697. [Google Scholar] [CrossRef] - Dąbrowski, A.; Kapitaniak, T. Using chaos to reduce oscillations: Experimental results. Chaos Solitons Fractals
**2009**, 39, 1677–1683. [Google Scholar] [CrossRef] - Dabrowski, A. The construction of the energy space. Chaos Solitons Fractals
**2005**, 26, 1277–1292. [Google Scholar] [CrossRef] - Dabrowski, A. New design of the impact damper. Mech. Mech. Eng.
**2000**, 4, 191–196. [Google Scholar] - Dabrowski, A.; Balcerzak, M.; Pikunov, D.; Stefanski, A. Improving efficiency of the largest Lyapunov exponent’s estimation by its determination from the vector field properties. Nonlinear Dyn.
**2020**, 102, 1869–1880. [Google Scholar] [CrossRef] - Parker, T.S.; Chua, L. Practical Numerical Algorithms for Chaotic Systems; Springer: New York, NY, USA, 1989. [Google Scholar]
- Dabrowski, A. The largest transversal Lyapunov exponent and master stability function from the perturbation vector and its derivative dot product (TLEVDP). Nonlinear Dyn.
**2012**, 69, 1225–1235. [Google Scholar] [CrossRef] [Green Version]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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