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Conservative chaos in a simple oscillatory system with non-smooth nonlinearity

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Abstract

In this paper, we consider some unusual features of dynamical regimes in the non-smooth potential \(V(x)=|x|\) which is a piece-wise linear function. Also, we consider the dynamics in more complicated potential \(V(x)=\left| |x|-a\right| \) which is quite similar to the well-known double-well potential within the Duffing model. Numerical results for Poincaré sections, bifurcation diagrams, and Lyapunov spectra together with dependencies of the largest Lyapunov characteristic exponent on the parameters of the excitation force are also obtained and analyzed. A comparison of the proposed systems and the Duffing model with the same fixed points is also done. Our numerical results show that such a relatively simple oscillatory system has rich nonlinear dynamics and exhibits a conservative character of chaos. This makes it possible to consider these systems as promising sources of chaotic signals in the field of modern chaos-based information technologies and digital communications.

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Notes

  1. Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points, with significantly different future trajectories. In other words, an arbitrarily small change of the current trajectory may lead to significantly different future behaviors.

  2. At first sight, it seems strange that changes in dynamics which are observable depend on such an unobservable variable as the phase. In reality, the dynamics of the system depends on the initial conditions which in turn determine an initial phase of the oscillatory motion. Thereby, changes in dynamics will depend on the relation between the initial phase of the oscillations and the initial phase of the excitation and we have no contradiction with classical results.

  3. To obtain the Lyapunov characteristic exponents for the Duffing potential, we consider an undamped Duffing equation

    $$\begin{aligned} {\ddot{x}}+\frac{\mathrm {d}V_{D}(x)}{\mathrm {d}x}=f(t),\quad f(t)=A\cos {\omega t},\quad V_{D}(x)=\alpha x^{4}+\beta x^{2}+\gamma \end{aligned}$$

    at the same initial conditions as for the double-well non-smooth potential.

References

  1. Aguilar-López, R., Martínez-Guerra, R., Perez-Pinacho, C.: Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. Eur. Phys. J. Spec. Top. 223(8), 1541–1548 (2014). https://doi.org/10.1140/epjst/e2014-02116-0

    Article  Google Scholar 

  2. Alemansour, H., Miandoab, E.M., Pishkenari, H.N.: Effect of size on the chaotic behavior of nano resonators. Commun. Nonlinear Sci. Numer. Simul. 44, 495–505 (2017). https://doi.org/10.1016/j.cnsns.2016.09.010

    Article  MathSciNet  MATH  Google Scholar 

  3. Alligood, K.T., Sauer, T., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer, New York (1996). https://doi.org/10.1007/b97589

    Book  MATH  Google Scholar 

  4. Armand Eyebe Fouda, J., Bodo, B., Djeufa, G.M., Sabat, S.L.: Experimental chaos detection in the Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 33, 259–269 (2016). https://doi.org/10.1016/j.cnsns.2015.09.011

    Article  MathSciNet  MATH  Google Scholar 

  5. Arneodo, A., Coullet, P.H., Spiegel, E.A.: Chaos in a finite macroscopic system. Phys. Lett. A 92(8), 369–373 (1982). https://doi.org/10.1016/0375-9601(82)90455-8

    Article  MathSciNet  Google Scholar 

  6. Arneodo, A., Coullet, P.H., Spiegel, E.A.: The dynamics of triple convection. Geophys. Astrophys. Fluid Dyn. 31(1–2), 1–48 (1985). https://doi.org/10.1080/03091928508219264

    Article  MATH  Google Scholar 

  7. Arneodo, A., Coullet, P.H., Spiegel, E.A., Tresser, C.: Asymptotic chaos. Physica D 14(3), 327–347 (1985). https://doi.org/10.1016/0167-2789(85)90093-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Arneodo, A., Coullet, P.H., Tresser, C.: Possible new strange attractors with spiral structure. Commun. Math. Phys. 79, 573–579 (1981). https://doi.org/10.1007/BF01209312

    Article  MathSciNet  MATH  Google Scholar 

  9. Awrejcewicz, J., Krysko, A.V., Erofeev, N.P., Dobriyan, V., Barulina, M.A., Krysko, V.A.: Quantifying chaos by various computational methods. Part 1: simple systems. Entropy 20(3), 175(1–28) (2018). https://doi.org/10.3390/e20030175

    Article  MathSciNet  Google Scholar 

  10. Awrejcewicz, J., Kudra, G., Lamarque, C.H.: Investigation of triple pendulum with impacts using fundamental solution matrices. Int. J. Bifurc. Chaos 14(12), 4191–4213 (2004). https://doi.org/10.1142/S0218127404011818

    Article  MathSciNet  MATH  Google Scholar 

  11. Awrejcewicz, J., Kudra, G., Wasilewski, G.: Chaotic dynamics of triple pendulum observed experimentally and numerically. In: XXII Symposium—Vibrations in Physical Systems—Poznan-Bedlewo 2006, pp. 53–58 (2006)

  12. Awrejcewicz, J., Kudra, G., Wasilewski, G.: Experimental and numerical investigation of chaotic regions in the triple physical pendulum. Nonlinear Dyn. 50(6), 755–766 (2007). https://doi.org/10.1007/s11071-007-9235-0

    Article  MATH  Google Scholar 

  13. Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, vol. 163. Springer, Dordrecht (2008). https://doi.org/10.1007/978-1-84628-708-4

    Book  MATH  Google Scholar 

  14. Blagojević, S.N., Čupić, Ž., Ivanović-Šašić, A., Kolar-Anić, L.: Mixed-mode oscillations and chaos in return maps of an oscillatory chemical reaction. Russ. J. Phys. Chem. A 89(13), 2349–2358 (2015). https://doi.org/10.1134/S0036024415130063

    Article  Google Scholar 

  15. Boeing, G.: Visual analysis of nonlinear dynamical systems: chaos, fractals, self-similarity and the limits of prediction. Systems 4(4), 37–54 (2016). https://doi.org/10.3390/systems4040037

    Article  Google Scholar 

  16. Budd, C.J.: Non-smooth dynamical systems and the grazing bifurcation. In: Aston, P. (ed.) Nonlinear Mathematics and its Applications, pp. 219–235. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  17. Cang, S., Li, Y., Xue, W., Wang, Z., Chen, Z.: Conservative chaos and invariant tori in the modified Sprott A system. Nonlinear Dyn. 99(2), 1699–1708 (2020). https://doi.org/10.1007/s11071-019-05385-9

    Article  Google Scholar 

  18. Cang, S., Wu, A., Zhang, R., Wang, Z., Chen, Z.: Conservative chaos in a class of nonconservative systems: theoretical analysis and numerical demonstrations. Int. J. Bifurc. Chaos 28(07), 1850087 (2018). https://doi.org/10.1142/S0218127418500876

    Article  MathSciNet  MATH  Google Scholar 

  19. Cvitanovic, P.: Universality in Chaos. Routledge, New York (2017)

    Google Scholar 

  20. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985). https://doi.org/10.1103/RevModPhys.57.617

    Article  MathSciNet  MATH  Google Scholar 

  21. Elyseeva, J.: On relative oscillation theory for symplectic eigenvalue problems. Appl. Math. Lett. 23(10), 1231–1237 (2010). https://doi.org/10.1016/j.aml.2010.06.004

    Article  MathSciNet  MATH  Google Scholar 

  22. Emelianova, Y.P., Kuznetsov, A.P., Sataev, I.R., Turukina, L.V.: Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Phys. D 244(1), 36–49 (2013). https://doi.org/10.1016/j.physd.2012.10.012

    Article  MATH  Google Scholar 

  23. Emelianova, Y.P., Kuznetsov, A.P., Turukina, L.V., Sataev, I.R., Chernyshov, N.Y.: A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1203–1212 (2014). https://doi.org/10.1016/j.cnsns.2013.08.004

    Article  MathSciNet  MATH  Google Scholar 

  24. Feldbrugge, J., Lehners, J.L., Turok, N.: No smooth beginning for spacetime. Phys. Rev. Lett. 119(17), 171301 (2017). https://doi.org/10.1007/BFb0103843

    Article  Google Scholar 

  25. Fiedler, R., Hetzler, H.: Numerical approximation of Lyapunov-exponents for quasiperiodic motions. MATEC Web Conf. 241, 01009(1–4) (2018). https://doi.org/10.1051/matecconf/201824101009

    Article  Google Scholar 

  26. Fu, S., Liu, Y., Ma, H., Du, Y.: Control chaos to different stable states for a piecewise linear circuit system by a simple linear control. Chaos, Sol. Fractals 130, 109431(1–8) (2020). https://doi.org/10.1016/j.chaos.2019.109431

    Article  MathSciNet  Google Scholar 

  27. Gendelman, O., Kravetc, P., Rachinskii, D.: Mixed global dynamics of forced vibro-impact oscillator with Coulomb friction. Chaos: Interdiscip. J. Nonlinear Sci. 29(11), 113116(1–11) (2019). https://doi.org/10.1063/1.5095627

    Article  MathSciNet  MATH  Google Scholar 

  28. Gesztesy, F., Zinchenko, M.: Renormalized oscillation theory for Hamiltonian systems. Adv. Math. 311, 569–597 (2017). https://doi.org/10.1016/j.aim.2017.03.005

    Article  MathSciNet  MATH  Google Scholar 

  29. Golbabai, A., Fardi, M., Sayevand, K.: Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system. Math. Comput. Modell. 58(11), 1837–1843 (2013). https://doi.org/10.1016/j.mcm.2011.12.027

    Article  MathSciNet  MATH  Google Scholar 

  30. Goldhirsch, I., Sulem, P.L., Orszag, S.A.: Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Phys. D 27(3), 311–337 (1987). https://doi.org/10.1016/0167-2789(87)90034-0

    Article  MathSciNet  MATH  Google Scholar 

  31. Gritli, H.: Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator. Chaos, Sol. Fractals 127, 127–145 (2019). https://doi.org/10.1016/j.chaos.2019.06.035

    Article  MathSciNet  Google Scholar 

  32. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42. Springer, New York (1983). https://doi.org/10.1007/978-1-4612-1140-2

    Book  MATH  Google Scholar 

  33. Jain, H., Ranjan, A., Gupta, K.: Analysis of chaos in double pendulum. In: 2013 6th International Conference on Emerging Trends in Engineering and Technology, pp. 171–176 (2013). https://doi.org/10.1109/ICETET.2013.50

  34. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In: Peitgen, H.O., Walther, H.O. (eds.) Functional Differential Equations and Approximation of Fixed Points, pp. 204–227. Springer, Berlin, Heidelberg (1979). https://doi.org/10.1007/BFb0064319

    Chapter  Google Scholar 

  35. Kautz, R.: Chaos. The Science of Predictable Random Motion. Oxford University Press, New York (2011)

    MATH  Google Scholar 

  36. Khlebodarova, T.M., Kogai, V.V., Fadeev, S.I., Likhoshvai, V.A.: Chaos and hyperchaos in simple gene network with negative feedback and time delays. J. Bioinf. Comput. Biol. 15(02), 1650042(1–19) (2017). https://doi.org/10.1142/S0219720016500426

    Article  Google Scholar 

  37. Kovačić, I., Brennan, M.J. (eds.): The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley, New York (2011). https://doi.org/10.1002/9780470977859

    Book  MATH  Google Scholar 

  38. Kunze, M.: Non-smooth Dynamical Systems. Lecture Notes in Mathematics, vol. 1744. Springer, Hoboken (2000)

    Google Scholar 

  39. Kyzioł, J., Okniński, A.: Van der Pol–Duffing oscillator: global view of metamorphoses of the amplitude profiles. Int. J. Non-Linear Mech. 116, 102–106 (2019). https://doi.org/10.1016/j.ijnonlinmec.2019.06.006

    Article  Google Scholar 

  40. Landau, L., Lifshitz, E.: Course of Theoretical Physics. Mechanics, vol. 1. Pergamon Press, Oxford (1960)

    Google Scholar 

  41. Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Berlin (2013)

    Google Scholar 

  42. Leine, R., Van Campen, D., Van de Vrande, B.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23(2), 105–164 (2000). https://doi.org/10.1023/A:1008384928636

    Article  MathSciNet  MATH  Google Scholar 

  43. Li, S., Ma, X., Bian, X., Lai, S.K., Zhang, W.: Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator. Nonlinear Dyn. 99(2), 1621–1642 (2020). https://doi.org/10.1007/s11071-019-05380-0

    Article  Google Scholar 

  44. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963). https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

    Article  MathSciNet  MATH  Google Scholar 

  45. Makarenkov, O., Lamb, J.S.: Dynamics and bifurcations of nonsmooth systems: A survey. Phys. D 241(22), 1826–1844 (2012). https://doi.org/10.1016/j.physd.2012.08.002

    Article  MathSciNet  Google Scholar 

  46. Maldacena, J., Shenker, S.H., Stanford, D.: A bound on chaos. J. High Energy Phys. 2016(8), 106(1–16) (2016). https://doi.org/10.1007/JHEP08(2016)106

    Article  MathSciNet  MATH  Google Scholar 

  47. Motallebzadeh, F., Motallebzadeh, F., Dadras, S., Ozgoli, S.: Controlling chaos in Arneodo system. In: 2009 17th Mediterranean Conference on Control and Automation, pp. 314–319 (2009). https://doi.org/10.1109/MED.2009.5164559

  48. Motter, A., Campbell, D.: Chaos at fifty. Phys. Today 66(5), 27–33 (2013). https://doi.org/10.1063/PT.3.1977

    Article  Google Scholar 

  49. Murillo-Escobar, M., Cruz-Hernández, C., Cardoza-Avendaño, L., Mendez-Ramirez, R.: A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 87(1), 407–425 (2017). https://doi.org/10.1007/s11071-016-3051-3

    Article  MathSciNet  Google Scholar 

  50. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2007). https://doi.org/10.1002/9783527617586

    Book  MATH  Google Scholar 

  51. Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/CBO9780511803260

    Book  MATH  Google Scholar 

  52. Paul Asir, M., Murali, K., Philominathan, P.: Strange nonchaotic attractors in oscillators sharing nonlinearity. Chaos, Sol. Fractals 118, 83–93 (2019). https://doi.org/10.1016/j.chaos.2018.11.016

    Article  MathSciNet  Google Scholar 

  53. Pei, J.S., Wright, J.P., Gay-Balmaz, F., Beck, J.L., Todd, M.D.: On choosing state variables for piecewise-smooth dynamical system simulations. Nonlinear Dyn. 95(2), 1165–1188 (2019). https://doi.org/10.1007/s11071-018-4622-2

    Article  MATH  Google Scholar 

  54. Prigogine, I., Stengers, I.: Order Out of Chaos: Man’s New Dialogue with Nature. Bantam Books, New York (1984)

    Google Scholar 

  55. Qi, G., Hu, J., Wang, Z.: Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos. Appl. Math. Model. 78, 350–365 (2020). https://doi.org/10.1016/j.apm.2019.08.023

    Article  MathSciNet  MATH  Google Scholar 

  56. Rasband, S.N.: Chaotic Dynamics of Nonlinear Systems. Wiley, New York (1990)

    MATH  Google Scholar 

  57. Salgado, R., Moore, H., Martens, J.W.M., Lively, T., Malik, S., McDermott, U., Michiels, S., Moscow, J.A., Tejpar, S., McKee, T., Lacombe, D.: Societal challenges of precision medicine: bringing order to chaos. Eur. J. Cancer 84, 325–334 (2017). https://doi.org/10.1016/j.ejca.2017.07.028

    Article  Google Scholar 

  58. Sciamanna, M., Shore, K.A.: Physics and applications of laser diode chaos. Nat. Photon. 9(3), 151–162 (2015). https://doi.org/10.1038/nphoton.2014.326

    Article  Google Scholar 

  59. Shinbrot, T., Grebogi, C., Wisdom, J., Yorke, J.A.: Chaos in a double pendulum. Am. J. Phys. 60(6), 491–499 (1992). https://doi.org/10.1119/1.16860

    Article  Google Scholar 

  60. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647–R650 (1994). https://doi.org/10.1103/PhysRevE.50.R647

    Article  MathSciNet  Google Scholar 

  61. Sprott, J.C.: Chaos and Time-Series Analysis. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  62. Sprott, J.C., Jafari, S., Pham, V.T., Hosseini, Z.S.: A chaotic system with a single unstable node. Phys. Lett. A 379(36), 2030–2036 (2015). https://doi.org/10.1016/j.physleta.2015.06.039

    Article  MathSciNet  MATH  Google Scholar 

  63. Strogatz, S.H.: Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Perseus Publishing, Cambridge (2000)

    MATH  Google Scholar 

  64. Syta, A., Litak, G., Lenci, S., Scheffler, M.: Chaotic vibrations of the Duffing system with fractional damping. Chaos: Interdiscip. J. Nonlinear Sci. 24(1), 013107 (2014). https://doi.org/10.1063/1.4861942

    Article  MathSciNet  MATH  Google Scholar 

  65. Vaidyanathan, S.: Output regulation of Arneodo–Coullet chaotic system. In: Meghanathan, N., Kaushik, B.K., Nagamalai, D. (eds.) Advanced Computing. CCSIT 2011. Communications in Computer and Information Science, pp. 98–107. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-17881-8_10

    Chapter  Google Scholar 

  66. Wang, M., Liao, X., Deng, Y., Li, Z., Su, Y., Zeng, Y.: Dynamics, synchronization and circuit implementation of a simple fractional-order chaotic system with hidden attractors. Chaos, Sol. Fractals 130, 109406(1–16) (2020). https://doi.org/10.1016/j.chaos.2019.109406

    Article  MathSciNet  Google Scholar 

  67. Wang, N., Zhang, G., Bao, H.: Infinitely many coexisting conservative flows in a 4d conservative system inspired by lc circuit. Nonlinear Dyn. 99, 1–20 (2020). https://doi.org/10.1007/s11071-020-05465-1

    Article  Google Scholar 

  68. Weiss, D., Küpper, T., Hosham, H.A.: Invariant manifolds for nonsmooth systems. Phys. D 241(22), 1895–1902 (2012). https://doi.org/10.1016/j.physd.2011.07.012

    Article  MathSciNet  Google Scholar 

  69. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16(3), 285–317 (1985). https://doi.org/10.1016/0167-2789(85)90011-9

    Article  MathSciNet  MATH  Google Scholar 

  70. Yadav, V.K., Shukla, V.K., Das, S.: Difference synchronization among three chaotic systems with exponential term and its chaos control. Chaos, Sol. Fractals 124, 36–51 (2019). https://doi.org/10.1016/j.chaos.2019.04.031

    Article  MathSciNet  Google Scholar 

  71. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  72. Zevin, A.: The theory of parametric oscillations. J. Appl. Math. Mech. 78(1), 30–38 (2014). https://doi.org/10.1016/j.jappmathmech.2014.05.004

    Article  MathSciNet  MATH  Google Scholar 

  73. Zhang, M., Yang, J.: Bifurcations and chaos in Duffing equation. Acta Math. Appl. Sin., Engl. Ser. 23, 665–684 (2007). https://doi.org/10.1007/s10255-007-0404

    Article  MathSciNet  MATH  Google Scholar 

  74. Zhang, Y., Yue, X., Du, L., Wang, L., Fang, T.: Generation and evolution of chaos in double-well Duffing oscillator under parametrical excitation. Shock Vib. 2016, 6109062(1–8) (2016). https://doi.org/10.1155/2016/6109062

    Article  Google Scholar 

  75. Zhu, Q., Ishitobi, M.: Experimental study of chaos in a driven triple pendulum. J. Sound Vib. 227(1), 230–238 (1999). https://doi.org/10.1006/jsvi.1999.2357

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous reviewers for their important comments and valuable suggestions which helped to improve the manuscript and made highlights clear.

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Authors and Affiliations

  1. Digital Technologies Department, Voronezh State University, Universitetskaya sq. 1, Voronezh, Russia, 394018

    Peter A. Meleshenko, Mikhail E. Semenov & Alexander F. Klinskikh

  2. Geophysical Survey of Russia Academy of Science, Lenina av. 189, Obninsk, Russia, 249035

    Mikhail E. Semenov

  3. Applied Mathematics and Mechanics Department, Voronezh State Technical University, XX-letiya Oktyabrya st. 84, Voronezh, Russia, 394006

    Mikhail E. Semenov

  4. Meteorology Department, Zhukovsky–Gagarin Air Force Academy, Starykh Bolshevikov st. 54 “A”, Voronezh, Russia, 394006

    Mikhail E. Semenov

  5. Target Search Lab of Groundbreaking Radio Communication Technologies of Advanced Research Foundation, Plekhanovskaya st. 14, Voronezh, Russia, 394018

    Peter A. Meleshenko

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This work was supported by the RFBR (Grants 18-08-00053-a and 19-08-00158). The contributions by M. E. Semenov and P. A. Meleshenko (Sections: Non-smooth potentials: some preliminaries; Numerical results: Poincaré sections, bifurcation diagrams, Lyapunov spectra) were supported by the RSF Grant No. 19-11-00197.

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Meleshenko, P.A., Semenov, M.E. & Klinskikh, A.F. Conservative chaos in a simple oscillatory system with non-smooth nonlinearity. Nonlinear Dyn 101, 2523–2540 (2020). https://doi.org/10.1007/s11071-020-05956-1

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