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Lyapunov exponent is a significant symbol to identify the nonlinear dynamic characteristics of the system. However, most of algorithms are not universal enough and complex. According to the classic Lyapunov exponent algorithm and perturbation theory, in this paper we propose a new algorithm which can be used to compute Lyapunov exponents for discontinuous systems. Firstly, the initial value of the system state parameter and the disturbance of each basic vector along the phase space are taken as initial conditions to determine the phase trajectory. Secondly, the method of difference quotient approximate derivative is adopted to obtain the Jacobi matrix. Thirdly, the eigenvalues of the Jacobi matrix are calculated to obtain the Lyapunov exponent spectrum of the system. Finally, the algorithm in a two-degree-of-freedom system with impacts and friction is used, showing its effectiveness and correctness by comparing its results with the counterparts from the synchronization method. The algorithm can not only be used for discrete systems and continuous-time dynamic systems, but also quickly calculate the Lyapunov exponent of complex discontinuous systems, which provides a new idea for determining the dynamic behavior of complex discontinuous systems.
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Keywords:
- Lyapunov exponent /
- discontinuous system /
- perturbation theory
[1] 刘秉正, 彭建华 2004 非线性动力学 (北京: 高等教育出版社) 第403−408页
Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) pp403−408 (in Chinese)
[2] Benettin G, Galgani L, Strelcyn J M 1976 Phys. Rev. A 14 2338
Google Scholar
[3] Shimada I, Nagashima T 1979 Prog. Theor. Phys. 61 1605
Google Scholar
[4] Oseledets V I 1968 Tr. Mosk. Mat. Obs. 19 179
[5] Benettin G, Galgani L, Giorgilli A, Strelcyn J M 1980 Meccanica 15 9
Google Scholar
[6] Benettin G, Galgani L, Giorgilli A, Strelcyn J M 1980 Meccanica 15 21
Google Scholar
[7] Wolf A 1986 Nonlinear Science: Theory and Applications (Manchester: Manchester University Press) pp273–290
[8] Parker T S, Chua L O 1989 Practical Numerical Algorithms for Chaotic Systems (Berlin: Springer-Verlag) pp73−81
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Google Scholar
[10] Balcerzak M, Pikunov D 2017 Mech. Eng. 21 985
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[15] Liao H T 2016 J. Comput. Phys. 313 57
Google Scholar
[16] Zhou S, Wang X Y, Wang Z, Zhang C 2019 Chaos 29 033125
Google Scholar
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Google Scholar
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Google Scholar
[19] Rosenstein M T, Collins J J, De Luca C J 1993 Physica D 65 117
Google Scholar
[20] Kantz H 1994 Phys. Lett. A 185 77
Google Scholar
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Google Scholar
Yang S Q, Zhang X H, Zhao C A 2000 Acta Phys. Sin. 49 636
Google Scholar
[22] 卢山, 王海燕 2006 55 572
Google Scholar
Lu S, Wang H Y 2006 Acta Phys. Sin. 55 572
Google Scholar
[23] 周双, 冯勇, 吴文渊, 汪维华 2016 65 020502
Google Scholar
Zhou S, Feng Y, Wu W Y, Wang W H 2016 Acta Phys. Sin. 65 020502
Google Scholar
[24] Pathak J, Lu Z X, Hunt B R, Girvan M, Ott E 2017 Chaos 27 121102
Google Scholar
[25] Zhou S, Wang X Y 2018 Chaos 28 123118
Google Scholar
[26] Shimizu T P, Takeuchi K A 2018 Chaos 28 121103
Google Scholar
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Google Scholar
[28] Stefański A, Kapitaniak T 2003 Chaos Solitons Fractals 15 233
Google Scholar
[29] Jin L, Lu Q S, Twizell E H 2006 J. Sound Vibr. 298 1019
Google Scholar
[30] Gritli H, Belghith S 2015 Chaos Solitons Fractals 81 172
Google Scholar
[31] 李清都, 郭建丽 2014 63 100501
Google Scholar
Li Q D, Guo J L 2014 Acta Phys. Sin. 63 100501
Google Scholar
[32] Stefański A, Kapitaniak T 2000 Discrete Dyn. Nat. Soc. 4 207
Google Scholar
[33] Stefański A 2000 Chaos Solitons Fractals 11 2443
Google Scholar
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Google Scholar
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Google Scholar
[37] Barreto Netto D M, Brandão A, Paiva A, Pacheco P M, Savi M A 2020 J Braz. Soc. Mech. Sci. Eng. 42 475
Google Scholar
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图 1 两个扰动量
$ \Delta {\boldsymbol{x}} $ ,$ \Delta {{\boldsymbol{x}}_0} $ 的正交归一化Figure 1. Orthonormalization of two perturbations
$\Delta {\boldsymbol{x}} , ~ \Delta {{\boldsymbol{x}}_0}$ .
图 2 矩阵
$ {{\boldsymbol{U}}_t}({\boldsymbol{x}}) $ 逼近算法示意图Figure 2. Illustration of the algorithm of the
$ {{\boldsymbol{U}}_t}({\boldsymbol{x}}) $ matrix approximation.
图 3 不连续系统Lyapunov指数算法的示意图
Figure 3. Illustration of the algorithm of discontinuous system Lyapunov exponent.
图 4 二自由度干摩擦冲击振荡器模型
Figure 4. Model of the two-degrees-of-freedom (2-DoF) mechanical oscillator with impacts and friction.
图 5 二自由度干摩擦冲击振荡器系统的分叉图和3个最大Lyapunov指数图 (a) 状态变量
$ {y_1} $ ; (b) 状态变量$ {y_2} $ ; (c) 状态变量$ {y_3} $ Figure 5. Bifurcation diagram and the corresponding diagram of the three largest Lyapunov exponents of the 2-DoF mechanical oscillator system with impacts and friction: (a) State variable
$ {y_1} $ ; (b) state variable$ {y_2} $ ; (c) state variable$ {y_3} $ . -
[1] 刘秉正, 彭建华 2004 非线性动力学 (北京: 高等教育出版社) 第403−408页
Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) pp403−408 (in Chinese)
[2] Benettin G, Galgani L, Strelcyn J M 1976 Phys. Rev. A 14 2338
Google Scholar
[3] Shimada I, Nagashima T 1979 Prog. Theor. Phys. 61 1605
Google Scholar
[4] Oseledets V I 1968 Tr. Mosk. Mat. Obs. 19 179
[5] Benettin G, Galgani L, Giorgilli A, Strelcyn J M 1980 Meccanica 15 9
Google Scholar
[6] Benettin G, Galgani L, Giorgilli A, Strelcyn J M 1980 Meccanica 15 21
Google Scholar
[7] Wolf A 1986 Nonlinear Science: Theory and Applications (Manchester: Manchester University Press) pp273–290
[8] Parker T S, Chua L O 1989 Practical Numerical Algorithms for Chaotic Systems (Berlin: Springer-Verlag) pp73−81
[9] Dabrowski A 2014 Nonlinear Dyn. 78 1601
Google Scholar
[10] Balcerzak M, Pikunov D 2017 Mech. Eng. 21 985
[11] Balcerzak M, Dabrowski A, Pikunov D 2018 Nonlinear Dyn. 94 3053
Google Scholar
[12] Dabrowski A, Balcerzak M, Pikunov D, Stefanski A 2020 Nonlinear Dyn. 102 1869
Google Scholar
[13] Zhou S, Wang X Y 2021 Physica A 563 125478
Google Scholar
[14] Hinrichs N, Oestreich M, Popp K 1998 J. Sound Vib. 216 435
Google Scholar
[15] Liao H T 2016 J. Comput. Phys. 313 57
Google Scholar
[16] Zhou S, Wang X Y, Wang Z, Zhang C 2019 Chaos 29 033125
Google Scholar
[17] Takens F 1981 Lect. Notes Math. 898 366
Google Scholar
[18] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
Google Scholar
[19] Rosenstein M T, Collins J J, De Luca C J 1993 Physica D 65 117
Google Scholar
[20] Kantz H 1994 Phys. Lett. A 185 77
Google Scholar
[21] 杨绍清, 章新华, 赵长安 2000 49 636
Google Scholar
Yang S Q, Zhang X H, Zhao C A 2000 Acta Phys. Sin. 49 636
Google Scholar
[22] 卢山, 王海燕 2006 55 572
Google Scholar
Lu S, Wang H Y 2006 Acta Phys. Sin. 55 572
Google Scholar
[23] 周双, 冯勇, 吴文渊, 汪维华 2016 65 020502
Google Scholar
Zhou S, Feng Y, Wu W Y, Wang W H 2016 Acta Phys. Sin. 65 020502
Google Scholar
[24] Pathak J, Lu Z X, Hunt B R, Girvan M, Ott E 2017 Chaos 27 121102
Google Scholar
[25] Zhou S, Wang X Y 2018 Chaos 28 123118
Google Scholar
[26] Shimizu T P, Takeuchi K A 2018 Chaos 28 121103
Google Scholar
[27] Krishnamurthy K, Manoharan S C, Swaminathan R 2020 J. Ambient Intell. Humaniz. Comput. 11 3329
Google Scholar
[28] Stefański A, Kapitaniak T 2003 Chaos Solitons Fractals 15 233
Google Scholar
[29] Jin L, Lu Q S, Twizell E H 2006 J. Sound Vibr. 298 1019
Google Scholar
[30] Gritli H, Belghith S 2015 Chaos Solitons Fractals 81 172
Google Scholar
[31] 李清都, 郭建丽 2014 63 100501
Google Scholar
Li Q D, Guo J L 2014 Acta Phys. Sin. 63 100501
Google Scholar
[32] Stefański A, Kapitaniak T 2000 Discrete Dyn. Nat. Soc. 4 207
Google Scholar
[33] Stefański A 2000 Chaos Solitons Fractals 11 2443
Google Scholar
[34] Stefański A, Dabrowski A, Kapitaniak T 2005 Chaos Solitons Fractals 23 1651
Google Scholar
[35] [36] Baumann M, Leine R I 2017 Procedia IUTAM 20 26
Google Scholar
[37] Barreto Netto D M, Brandão A, Paiva A, Pacheco P M, Savi M A 2020 J Braz. Soc. Mech. Sci. Eng. 42 475
Google Scholar
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