SpringerLink
Log in

Chaotic behaviors and multiple attractors in a double pendulum with an external harmonic excitation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the dynamic behavior of a double pendulum under horizontal harmonic excitation is studied. The mathematical model is described by a four-dimensional non-autonomous system with smooth nonlinearities. Based on the sensitivity analysis of parameters, two representative parameters are selected and their influences on the system behavior are reported with a set of high-resolution stability diagrams. In addition to explore the classical dynamic behavior of the system, our study also investigates the issue of multistability arising from attractor self-reproducing. To enhance the reliability of our findings, simulations were conducted within a multi-body simulation environment, which yielded consistent and robust results. Furthermore, utilizing the experimental platform developed with Qualisys, we identified several pairs of attractors with specific offsets, a significant indication of attractor self-reproducing. This paper will contribute to understand the rich and intriguing behaviors of the double pendulum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Suzuki, Y., Nomura, T., Casadio, M., Morasso, P.: Intermittent control with ankle, hip, and mixed strategies during quiet standing: a theoretical proposal based on a double inverted pendulum model. J. Theor. Biol. 310, 55–79 (2012)

    MathSciNet  Google Scholar 

  2. Liu, C., Ning, J., Chen, Q.: Dynamic walking control of humanoid robots combining linear inverted pendulum mode with parameter optimization. Int. J. Adv. Robot. Syst. 15(1), 1729881417749672 (2018)

    Google Scholar 

  3. Narukawa, T., Takahashi, M., Yoshida, K.: Biped locomotion on level ground by torso and swing-leg control based on passive-dynamic walking. In: 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, pp. 4009–4014 (2005)

  4. Fathizadeh, M., Mohammadi, H., Taghvaei, S.: A modified passive walking biped model with two feasible switching patterns of motion to resemble multi-pattern human walking. Chaos Solitons Fractals 127, 83–95 (2019)

    MathSciNet  Google Scholar 

  5. Gritli, H., Khraief, N., Belghith, S.: Chaos control in passive walking dynamics of a compass-gait model. Commun. Nonlinear Sci. Numer. Simul. 18(8), 2048–2065 (2013)

    MathSciNet  Google Scholar 

  6. Li, Q., Guo, J., Yang, X.-S.: Bifurcation and chaos in the simple passive dynamic walking model with upper body. Chaos: Interdiscip. J. Nonlinear Sci. 24(3), 033114 (2014)

    MathSciNet  Google Scholar 

  7. Vasileiou, C., Smyrli, A., Drogosis, A., Papadopoulos, E.: Development of a passive biped robot digital twin using analysis, experiments, and a multibody simulation environment. Mech. Mach. Theory 163, 104346 (2021)

    Google Scholar 

  8. Afshar, P.N., Ren, L.: Dynamic stability of passive bipedal walking on rough terrain: a preliminary simulation study. J. Bionic Eng. 9(4), 423–433 (2012)

    Google Scholar 

  9. Gritli, H., Belghith, S., Khraeif, N.: Intermittency and interior crisis as route to chaos in dynamic walking of two biped robots. Int. J. Bifurc. Chaos 22(03), 1250056 (2012)

    Google Scholar 

  10. Asai, Y., Tasaka, Y., Nomura, K., Nomura, T., Casadio, M., Morasso, P.: A model of postural control in quiet standing: robust compensation of delay-induced instability using intermittent activation of feedback control. PLoS ONE 4(7), e6169 (2009)

    Google Scholar 

  11. Dudkowski, D., Wojewoda, J., Czołczyński, K., Kapitaniak, T.: Is it really chaos? The complexity of transient dynamics of double pendula. Nonlinear Dyn. 102, 759–770 (2020)

    Google Scholar 

  12. Kovacic, I., Zukovic, M., Radomirovic, D.: Normal modes of a double pendulum at low energy levels. Nonlinear Dyn. 99(3), 1893–1908 (2020)

    Google Scholar 

  13. Collins, S.H., Wisse, M., Ruina, A.: A three-dimensional passive-dynamic walking robot with two legs and knees. Int. J. Robot. Res. 20(7), 607–615 (2001)

    Google Scholar 

  14. Litak, G., Margielewicz, J., Gkaska, D., Yurchenko, D., Dkabek, K.: Dynamic response of the spherical pendulum subjected to horizontal Lissajous excitation. Nonlinear Dyn. 102(4), 2125–2142 (2020)

    Google Scholar 

  15. Kudra, G., Szewc, M., Ludwicki, M., Awrejcewicz, J.: Modeling and simulation of bifurcation dynamics of a double spatial pendulum excited by a rotating obstacle. Int. J. Struct. Stab. Dyn. 19(12), 1950145 (2019)

    MathSciNet  Google Scholar 

  16. Roy, J., Mallik, A.K., Bhattacharjee, J.K.: Role of initial conditions in the dynamics of a double pendulum at low energies. Nonlinear Dyn. 73(1), 993–1004 (2013)

    MathSciNet  Google Scholar 

  17. Shinbrot, T., Grebogi, C., Wisdom, J., Yorke, J.A.: Chaos in a double pendulum. Am. J. Phys. 60(6), 491–499 (1992)

    Google Scholar 

  18. Yu, P., Bi, Q.: Analysis of non-linear dynamics and bifurcations of a double pendulum. J. Sound Vib. 217(4), 691–736 (1998)

    MathSciNet  Google Scholar 

  19. Skeldon, A.: Dynamics of a parametrically excited double pendulum. Physica D 75(4), 541–558 (1994)

    MathSciNet  Google Scholar 

  20. Rafat, M., Wheatland, M., Bedding, T.: Dynamics of a double pendulum with distributed mass. Am. J. Phys. 77(3), 216–223 (2009)

    Google Scholar 

  21. Stachowiak, T., Okada, T.: A numerical analysis of chaos in the double pendulum. Chaos Solitons Fractals 29(2), 417–422 (2006)

    Google Scholar 

  22. Marszal, M., Jankowski, K., Perlikowski, P., Kapitaniak, T.: Bifurcations of oscillatory and rotational solutions of double pendulum with parametric vertical excitation. Math. Probl. Eng. 2014, 892793 (2014)

    MathSciNet  Google Scholar 

  23. Lampart, M., Zapoměl, J.: Dynamics of a non-autonomous double pendulum model forced by biharmonic excitation with soft stops. Nonlinear Dyn. 99(3), 1909–1921 (2020)

    Google Scholar 

  24. Kumar, R., Gupta, S., Ali, S.F.: Energy harvesting from chaos in base excited double pendulum. Mech. Syst. Signal Process. 124, 49–64 (2019)

    Google Scholar 

  25. Freire, J.G., Meucci, R., Arecchi, F.T., Gallas, J.A.: Self-organization of pulsing and bursting in a CO2 laser with opto-electronic feedback. Chaos: Interdiscip. J. Nonlinear Sci. 25(9), 097607 (2015)

    Google Scholar 

  26. Rao, X., Zhao, X., Chu, Y., Zhang, J., Gao, J.: The analysis of mode-locking topology in an sir epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees. Chaos Solitons Fractals 139, 110031 (2020)

    MathSciNet  Google Scholar 

  27. Fazanaro, F.I., Soriano, D.C., Suyama, R., Madrid, M.K., de Oliveira, J.R., Muñoz, I.B., Attux, R.: Numerical characterization of nonlinear dynamical systems using parallel computing: the role of GPUs approach. Commun. Nonlinear Sci. Numer. Simul. 37, 143–162 (2016)

    MathSciNet  Google Scholar 

  28. MSC ADAMS: Automatic Dynamic Analysis of Mechanical Systems. MSC Software Corporation, Newport Beach (2014)

    Google Scholar 

  29. Liu, B., Hu, H.: Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum. Int. J. Non-Linear Mech. 45(4), 442–452 (2010)

    Google Scholar 

  30. Minguzzi, E.: Rayleigh’s dissipation function at work. Eur. J. Phys. 36(3), 035014 (2015)

    MathSciNet  Google Scholar 

  31. Alexander, R.: Solving ordinary differential equations I: nonstiff problems (E. Hairer, S. P. Norsett, and G. Wanner). SIAM Rev. 32(3), 485 (1990)

    Google Scholar 

  32. Press, W.H.: Numerical Recipes 3rd edition: The Art of Scientific Computing. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  33. Microsoft Corporation: Microsoft Visual C++. Microsoft Corporation, Redmond (1995)

    Google Scholar 

  34. Symbolic Math Toolbox, et al.: Matlab. Mathworks Inc (1993)

  35. Laufenberg, M.J., Pai, M.: A new approach to dynamic security assessment using trajectory sensitivities. In: Proceedings of the 20th International Conference on Power Industry Computer Applications, IEEE, pp. 272–277 (1997)

  36. Lai, X., Wang, S., Ma, S., **e, J., Zheng, Y.: Parameter sensitivity analysis and simplification of equivalent circuit model for the state of charge of lithium-ion batteries. Electrochim. Acta 330, 135239 (2020)

    Google Scholar 

  37. Khalil, H.: Nonlinear Systems. Prentice Hall, Pearson Education, New York (2002)

    Google Scholar 

  38. Freire, J.G., Gallas, J.A.: Stern-Brocot trees in the periodicity of mixed-mode oscillations. Phys. Chem. Chem. Phys. 13(26), 12191–12198 (2011)

    Google Scholar 

  39. Cabeza, C., Briozzo, C.A., Garcia, R., Freire, J.G., Marti, A.C., Gallas, J.A.: Periodicity hubs and wide spirals in a two-component autonomous electronic circuit. Chaos Solitons Fractals 52, 59–65 (2013)

    MathSciNet  Google Scholar 

  40. Rao, X., Zhao, X., Gao, J., Zhang, J.: Self-organizations with fast-slow time scale in a memristor-based Shinriki’s circuit. Commun. Nonlinear Sci. Numer. Simul. 94, 105569 (2021)

    MathSciNet  Google Scholar 

  41. Rao, X., Chu, Y., Zhang, J., Gao, J.: Complex mode-locking oscillations and Stern-Brocot derivation tree in a CSTR reaction with impulsive perturbations. Chaos: Interdiscip. J. Nonlinear Sci. 30(11), 113117 (2020)

    MathSciNet  Google Scholar 

  42. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)

    MathSciNet  Google Scholar 

  43. Lampart, M., Zapoměl, J.: The disturbance influence on vibration of a belt device driven by a crank mechanism. Chaos Solitons Fractals 173, 113634 (2023)

    MathSciNet  Google Scholar 

  44. Fazanaro, F.I., Soriano, D.C., Suyama, R., Madrid, M.K., de Oliveira, J.R., Muñoz, I.B., Attux, R.: Numerical characterization of nonlinear dynamical systems using parallel computing: the role of GPUs approach. Commun. Nonlinear Sci. Numer. Simul. 37, 143–162 (2016)

    MathSciNet  Google Scholar 

  45. Li, C., Sprott, J.C., Hu, W., Xu, Y.: Infinite multistability in a self-reproducing chaotic system. Int. J. Bifurc. Chaos 27(10), 1750160 (2017)

    MathSciNet  Google Scholar 

  46. Ramamoorthy, R., Rajagopal, K., Leutcho, G.D., Krejcar, O., Namazi, H., Hussain, I.: Multistable dynamics and control of a new 4D memristive chaotic Sprott B system. Chaos Solitons Fractals 156, 111834 (2022)

    Google Scholar 

  47. Li, C., Sprott, J.C., Zhang, X., Chai, L., Liu, Z.: Constructing conditional symmetry in symmetric chaotic systems. Chaos Solitons Fractals 155, 111723 (2022)

  48. Talla, F.C., Tchitnga, R., Fotso, P.L., Kengne, R., Nana, B., Fomethe, A.: Unexpected behaviors in a single mesh Josephson junction based self-reproducing autonomous system. Int. J. Bifurc. Chaos 30(07), 2050097 (2020)

    MathSciNet  Google Scholar 

  49. Li, C., Xu, Y., Chen, G., Liu, Y., Zheng, J.: Conditional symmetry: bond for attractor growing. Nonlinear Dyn. 95, 1245–1256 (2019)

    Google Scholar 

Download references

Acknowledgements

Authors acknowledge the financial support from the National Natural Science Foundation of China (51906225) and the Postdoctoral Research Sponsorship in Henan Province under (Grant no. 1902007).

Funding

The National Natural Science Foundation of China (51906225) and the Postdoctoral Research Sponsorship in Henan Province under (Grant no. 1902007).

Author information

Authors and Affiliations

  1. School of Mechanical and Power Engineering, Zhengzhou University, Science Road 100, Zhengzhou, 450001, China

    Zeyi Liu, Jianshe Gao, Shunliang Ding & **aobo Rao

Authors
  1. Zeyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianshe Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shunliang Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. **aobo Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **aobo Rao.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interest to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Gao, J., Ding, S. et al. Chaotic behaviors and multiple attractors in a double pendulum with an external harmonic excitation. Nonlinear Dyn 112, 1779–1796 (2024). https://doi.org/10.1007/s11071-023-09140-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09140-z

Keywords

Search

Navigation