An analytic estimation for the largest Lyapunov exponent of the Rössler chaotic system based on the synchronization method

Bin Zhen; Wenwen Liu; Lijun Pei; Bin Zhen; Wenwen Liu; Lijun Pei

doi:10.3934/era.2024120

Electronic Research Archive

2024, Volume 32, Issue 4: 2642-2664. doi: 10.3934/era.2024120

Previous Article Next Article

Research article Special Issues

An analytic estimation for the largest Lyapunov exponent of the Rössler chaotic system based on the synchronization method

1.
School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Received: 12 December 2023 Revised: 13 March 2024 Accepted: 26 March 2024 Published: 01 April 2024

In this paper, we propose an analytical approach to estimate the largest Lyapunov exponent (LLE) of a Rössler chaotic system, leveraging the synchronization method. This research focuses on establishing an analytical criterion for the synchronization of two identical Rössler chaotic systems through the linear coupling of state variables. This is crucial because the LLE of such systems can be estimated based on the critical coupling required for synchronization. Unlike previous studies, we first transform the synchronization error system between two identical Rössler chaotic systems into a set of Volterra integral equations by using the Laplace transform and convolution theorem. The critical coupling for synchronization is analytically derived using integral equation theory to solve the error system. As compared to the numerical results of the Rössler chaotic system's LLE, our analytical estimates demonstrate high accuracy. Our findings suggest that the challenge of estimating the Rössler chaotic system's LLE can be simplified to solving a cubic algebraic equation, offering a novel perspective on the analysis of how parameters influence the LLE's value in the Rössler chaotic system.
- largest Lyapunov exponent,
- Rössler system,
- chaos synchronization,
- analytic estimation,
- Volterra integral equations
Citation: Bin Zhen, Wenwen Liu, Lijun Pei. An analytic estimation for the largest Lyapunov exponent of the Rössler chaotic system based on the synchronization method[J]. Electronic Research Archive, 2024, 32(4): 2642-2664. doi: 10.3934/era.2024120

Related Papers:

Abstract

In this paper, we propose an analytical approach to estimate the largest Lyapunov exponent (LLE) of a Rössler chaotic system, leveraging the synchronization method. This research focuses on establishing an analytical criterion for the synchronization of two identical Rössler chaotic systems through the linear coupling of state variables. This is crucial because the LLE of such systems can be estimated based on the critical coupling required for synchronization. Unlike previous studies, we first transform the synchronization error system between two identical Rössler chaotic systems into a set of Volterra integral equations by using the Laplace transform and convolution theorem. The critical coupling for synchronization is analytically derived using integral equation theory to solve the error system. As compared to the numerical results of the Rössler chaotic system's LLE, our analytical estimates demonstrate high accuracy. Our findings suggest that the challenge of estimating the Rössler chaotic system's LLE can be simplified to solving a cubic algebraic equation, offering a novel perspective on the analysis of how parameters influence the LLE's value in the Rössler chaotic system.

References

[1]	V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems, (1968), 197–231. Available from: https://api.semanticscholar.org/CorpusID: 117573994.
[2]	J. L. Kaplan, J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, Springer, Berlin, (1979), 204–227. https://doi.org/10.1007/BFb0064319
[3]	G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15 (1980), 9–20. https://doi.org/10.1007/BF02128236 doi: 10.1007/BF02128236
[4]	E. N. Lorenz, The local structure of a chaotic attractor in four dimensions, Physica D, 13 (1984), 90–104. https://doi.org/10.1016/0167-2789(84)90272-0 doi: 10.1016/0167-2789(84)90272-0
[5]	S. Habib, R. D. Ryne, Symplectic calculation of Lyapunov exponents, Phys. Rev. Lett., 74 (1995), 70. https://doi.org/10.1103/PhysRevLett.74.70 doi: 10.1103/PhysRevLett.74.70
[6]	R. Franzosi, R. Gatto, G. Pettini, M. Pettini, Analytic Lyapunov exponents in a classical nonlinear field equation, Phys. Rev. E: Stat. Nonlinear Biol. Soft Matter Phys., 61 (2000), R3299. https://doi.org/10.1103/PhysRevE.61.R3299 doi: 10.1103/PhysRevE.61.R3299
[7]	R. Caponetto, S. Fazzino, A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 22–27. https://doi.org/10.1016/j.cnsns.2012.06.013 doi: 10.1016/j.cnsns.2012.06.013
[8]	A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9
[9]	P. Bryant, R. Brown, H. D. Abarbanel, Lyapunov exponents from observed time series, Phys. Rev. Lett., 65 (1990), 1523. https://doi.org/10.1103/PhysRevLett.65.1523 doi: 10.1103/PhysRevLett.65.1523
[10]	X. Zeng, R. Eykholt, R. Pielke, Estimating the Lyapunov-exponent spectrum from short time series of low precision, Phys. Rev. Lett., 66 (1991), 3229. https://doi.org/10.1103/PhysRevLett.66.3229 doi: 10.1103/PhysRevLett.66.3229
[11]	Y. Perederiy, Method for calculation of Lyapunov exponents spectrum from data series, Izvestiya VUZ. Appl. Nonlinear Dyn., 20 (2012), 99–104.
[12]	K. Briggs, An improved method for estimating Liapunov exponents of chaotic time series, Phys. Lett. A, 151 (1990), 27–32. https://doi.org/10.1016/0375-9601(90)90841-B doi: 10.1016/0375-9601(90)90841-B
[13]	H. F. von Bremen, F. E. Udwadia, W. Proskurowski, An efficient QR based method for the computation of Lyapunov exponents, Physica D, 101 (1997), 1–16. https://doi.org/10.1016/S0167-2789(96)00216-3 doi: 10.1016/S0167-2789(96)00216-3
[14]	L. Dieci, C. Elia, SVD algorithms to approximate spectra of dynamical systems, Math. Comput. Simul., 79 (2008), 1235–1254. https://doi.org/10.1016/j.matcom.2008.03.005 doi: 10.1016/j.matcom.2008.03.005
[15]	A. Dabrowski, Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product, Nonlinear Dyn., 67 (2012), 283–291. https://doi.org/10.1007/s11071-011-9977-6 doi: 10.1007/s11071-011-9977-6
[16]	H. Liao, Novel gradient calculation method for the largest Lyapunov exponent of chaotic systems, Nonlinear Dyn., 85 (2016), 1377–1392. https://doi.org/10.1007/s11071-016-2766-5 doi: 10.1007/s11071-016-2766-5
[17]	L. Escot, J. E. Sandubete, Estimating Lyapunov exponents on a noisy environment by global and local Jacobian indirect algorithms, Appl. Math. Comput., 436 (2023), 127498. https://doi.org/10.1016/j.amc.2022.127498 doi: 10.1016/j.amc.2022.127498
[18]	S. Zhou, X. Y. Wang, Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations, Chaos, Solitons Fractals, 139 (2020), 109981. https://doi.org/10.1016/j.chaos.2020.109981 doi: 10.1016/j.chaos.2020.109981
[19]	J. He, S. Yu, J. Cai, Numerical analysis and improved algorithms for Lyapunov-exponent calculation of discrete-time chaotic systems, Int. J. Bifurcation Chaos, 26 (2016), 1650219. https://doi.org/10.1142/S0218127416502199 doi: 10.1142/S0218127416502199
[20]	S. Zhou, X. Wang, Z. Wang, C. Zhang, A novel method based on the pseudo-orbits to calculate the largest Lyapunov exponent from chaotic equations, Chaos, 29 (2019), 033125. https://doi.org/10.1063/1.5087512 doi: 10.1063/1.5087512
[21]	J. Pathak, Z. X. Lu, B. R. Hunt, M. Cirvan, E. Ott, Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data, Chaos, 27 (2017), 121102. https://doi.org/10.1063/1.5010300 doi: 10.1063/1.5010300
[22]	A. McAllister, M. McCartney, D. H. Glass, Stability, collapse and hyperchaos in a class of tri-trophic predator-prey models, Physica A, 628 (2023), 129146. https://doi.org/10.1016/j.physa.2023.129146 doi: 10.1016/j.physa.2023.129146
[23]	M. Balcerzak, A. Dabrowski, O. B. Blazejczyk, A. Stefanski, Determining Lyapunov exponents of non-smooth systems: perturbation vectors approach, Mech. Syst. Signal Process., 141 (2020), 106734. https://doi.org/10.1016/j.ymssp.2020.106734 doi: 10.1016/j.ymssp.2020.106734
[24]	D. C. Soriano, F. I. Fazanaro, R. Suyama, J. R. de Oliveira, R. Attux, M. K. Madrid, A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh-Nagumo model, Nonlinear Dyn., 67 (2012), 413–424. https://doi.org/10.1007/s11071-011-9989-2 doi: 10.1007/s11071-011-9989-2
[25]	A. Stefanski, T. Kapitaniak, Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems, Discrete Dyn. Nat. Soc., 4 (1999), 207–215. https://doi.org/10.1155/S1026022600000200 doi: 10.1155/S1026022600000200
[26]	A. Stefanski, Estimation of the largest Lyapunov exponent in systems with impacts, Chaos, Solitons Fractals, 11 (2000), 2443–2451. https://doi.org/10.1016/S0960-0779(00)00029-1 doi: 10.1016/S0960-0779(00)00029-1
[27]	B. Kharabian, H. Mirinejad, Synchronization of Rossler chaotic systems via hybrid adaptive backstepping/sliding mode control, Results Control Optim., 4 (2021), 100020. https://doi.org/10.1016/j.rico.2021.100020 doi: 10.1016/j.rico.2021.100020
[28]	B. Kharabian, H. Mirinejad, Fuzzy Lyapunov exponents placement for chaos stabilization, Physica D, 445 (2023), 133648. https://doi.org/10.1016/j.physd.2023.133648 doi: 10.1016/j.physd.2023.133648
[29]	J. Ritt, Evaluation of entrainment of a nonlinear neural oscillator to white noise, Phys. Rev. E: Stat. Nonlinear Biol. Soft Matter Phys., 68 (2003), 041915. https://doi.org/10.1103/PhysRevE.68.041915 doi: 10.1103/PhysRevE.68.041915
[30]	K. Pakdaman, D. Mestivier, Noise induced synchronization in a neuronal oscillator, Physica D, 192 (2004), 123–137. https://doi.org/10.1016/j.physd.2003.12.006 doi: 10.1016/j.physd.2003.12.006
[31]	D. S. Goldobin, A. S. Pikovsky, Antireliability of noise-driven neurons, Phys. Rev. E: Stat. Nonlinear Biol. Soft Matter Phys., 73 (2006), 061906. https://doi.org/10.1103/PhysRevE.73.061906 doi: 10.1103/PhysRevE.73.061906
[32]	J. N. Teramae, D. Tanaka, Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators, Phys. Rev. Lett., 93 (2004), 204103. https://doi.org/10.1103/PhysRevLett.93.204103 doi: 10.1103/PhysRevLett.93.204103
[33]	D. S. Goldobin, A. S. Pikovsky, Synchronization of self-sustained oscillators by common white noise, Physica A, 351 (2005), 126–132. https://doi.org/10.1016/j.physa.2004.12.014 doi: 10.1016/j.physa.2004.12.014
[34]	D. S. Goldobin, J. N. Teramae, H. Nakao, G. B. Ermentrout, Dynamics of limit-cycle oscillators subject to general noise, Phys. Rev. Lett., 105 (2010), 154101. https://doi.org/10.1103/PhysRevLett.105.154101 doi: 10.1103/PhysRevLett.105.154101
[35]	A. E. Hramov, A. A. Koronovskii, M. K. Kurovskaya, O. I. Moskalenko, Analytical expression for zero Lyapunov exponent of chaotic noised oscillators, Chaos, Solitons Fractals, 78 (2015), 118–123. https://doi.org/10.1016/j.chaos.2015.07.016 doi: 10.1016/j.chaos.2015.07.016
[36]	A. E. Hramov, A. A. Koronovskii, M. K. Kurovskaya, Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise, Phys. Rev. E: Stat. Nonlinear Biol. Soft Matter Phys., 78 (2008), 036212. https://doi.org/10.1103/PhysRevE.78.036212 doi: 10.1103/PhysRevE.78.036212
[37]	A. Politi, F. Ginelli, S. Yanchuk, Y. Maistrenko, From synchronization to Lyapunov exponents and back, Physica D, 224 (2006), 90–101. https://doi.org/10.1016/j.physd.2006.09.032 doi: 10.1016/j.physd.2006.09.032
[38]	O. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1976), 397–398. https://doi.org/10.1016/0375-9601(76)90101-8 doi: 10.1016/0375-9601(76)90101-8
[39]	H. Fujisaka, T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Prog. Theor. Phys., 69 (1983), 32–47. https://doi.org/10.1143/PTP.69.32 doi: 10.1143/PTP.69.32
[40]	L. M. Pecora, T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109. https://doi.org/10.1103/PhysRevLett.80.2109 doi: 10.1103/PhysRevLett.80.2109
[41]	G. P. Jiang, W. K. S. Tang, A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach, Int. J. Bifurcation Chaos, 12 (2002), 2239–2253. https://doi.org/10.1142/S0218127402005790 doi: 10.1142/S0218127402005790
[42]	L. Kocarev, U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett., 74 (1995), 5028–5031. https://doi.org/10.1103/PhysRevLett.74.5028 doi: 10.1103/PhysRevLett.74.5028
[43]	L. O. Chua, L. Kocarev, K. Eckert, M. Itoh, Experimental chaos synchronization in Chua's circuit, Int. J. Bifurcation Chaos, 2 (1992), 705–708. https://doi.org/10.1142/S0218127492000811 doi: 10.1142/S0218127492000811
[44]	J. A. Nohel, Some problems in nonlinear Volterra integral equations, Bull. Amer. Math. Soc., 68 (1962), 323–329. https://doi.org/10.1090/S0002-9904-1962-10790-3 doi: 10.1090/S0002-9904-1962-10790-3
[45]	L. C. P. Marcia, G. N. Erivelton, A. M. M. Samir, J. L. Marcio, Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm, Chaos, Solitons Fractals, 112 (2018), 36–43. https://doi.org/10.1016/j.chaos.2018.04.032 doi: 10.1016/j.chaos.2018.04.032

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)