Skip to main content
SpringerLink
Log in

Nonlinear dynamics of a non-autonomous network with coupled discrete–continuum oscillators

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

Abstract

A network model of a multi-modular floating platform incorporated with a runway structure, viewed as a non-autonomous network with discrete–continuum oscillators, is developed for a general purpose of dynamic analysis. Numerical analysis shows the coupling effect between the two different types of oscillators on various complex dynamics, including sudden leaps, torus motions, beating vibrations, the synergetic effect of phase lock and anti-phase synchronizations. The amplitude death phenomenon, a suppressed weak oscillation state, is studied by using the fundamental solution derived by the averaging method. The parametric domain of the onset of amplitude death is illustrated to show the great significance to the stability design of the floating platform. The effect of the flexural rigidity of the runway on the distribution of amplitude death state is also discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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

Similar content being viewed by others

References

  1. Kuznetsov, A.P., Stankevich, N.V., Tyuryukina, L.V.: The death of quasi-periodic regimes in a system of dissipatively coupled van der Pol oscillators under pulsed drive action. Tech. Phys. Lett. 34, 643–645 (2008)

    Google Scholar 

  2. Bar-Eli, K.: Coupling of chemical oscillators. J. Phys. Chem. 88, 3616–3622 (1984)

    Article  Google Scholar 

  3. Hastings, A.: Transient dynamics and persistence of ecological systems. Ecol. Lett. 4, 215–220 (2001)

    Article  Google Scholar 

  4. Li, J., Yu, J., Miao, Z., Zhou, J.: Region reaching control of networked robot systems. In: Proceedings of the 36th Chinese Control Conference, pp. 8662–8667, Dalian, China (2017)

  5. Yu, J., Ji, J., Miao, Z., Zhou, J.: Adaptive formation control of networked Lagrangian systems with a moving leader. Nonlinear Dyn. 90, 2755–2766 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  7. Kaneko, K.: Theory and Applications of Coupled Map Lattices. Wiley, New York (1993)

    MATH  Google Scholar 

  8. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  9. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rosa, E., Ott, E., Hess, M.H.: Transition to phase synchronization of chaos. Phys. Rev. Lett. 80, 1642–1645 (1998)

    Article  Google Scholar 

  11. Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196 (1997)

    Article  MATH  Google Scholar 

  12. Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819 (1996)

    Article  Google Scholar 

  13. Zaks, M.A., Park, E.H., Rosenblum, M.G., Kurths, J.: Alternating locking ratios in imperfect phase synchronization. Phys. Rev. Lett. 82, 4228–4231 (1999)

    Article  Google Scholar 

  14. Boccaletti, S., Valladares, D.L.: Characterization of intermittent lag synchronization. Phys. Rev. E 62, 7497–7500 (2000)

    Article  Google Scholar 

  15. Femat, R., Solıs-Perales, G.: On the chaos synchronization phenomena. Phys. Lett. A 262, 50–60 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)

    Article  Google Scholar 

  17. Kiss, I.Z.: Emerging coherence in a population of chemical oscillators. Science 296, 1676–1678 (2002)

    Article  Google Scholar 

  18. Boccaletti, S., Bianconi, G., Criado, R., del Genio, C.I., Gómez-Gardeñes, J., Romance, M., Sendiña-Nadal, I., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014)

    Article  MathSciNet  Google Scholar 

  19. Matthews, P., Strogatz, S.: Phase diagram for the collective behavior of limit-cycle oscillators. Phys. Rev. Lett. 65, 1701–1704 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Resmi, V., Ambika, G., Amritkar, R.E.: Synchronized states in chaotic systems coupled indirectly through a dynamic environment. Phys. Rev. E 81, 46216 (2010)

    Article  Google Scholar 

  21. Zhang, X., Wu, Y., Peng, J.: Analytical conditions for amplitude death induced by conjugate variable couplings. Int. J. Bifurc. Chaos. 21, 225–235 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ermentrout, G.B.: Oscillator death in populations of “all to all” coupled nonlinear oscillators. Phys. D Nonlinear Phenom. 41, 219–231 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Aronson, D.G., Ermentrout, G.B., Kopell, N.: Amplitude response of coupled oscillators. Phys. D Nonlinear Phenom. 41, 403–449 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mirollo, R.E., Strogatz, S.H.: Amplitude death in an array of limit-cycle oscillators. J. Stat. Phys. 60, 245–262 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ramana Reddy, D., Sen, A., Johnston, G.: Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 80, 5109–5112 (1998)

    Article  Google Scholar 

  26. Reddy, D.V.R., Sen, A., Johnston, G.L.: Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Phys. D Nonlinear Phenom. 129, 15–34 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ramana Reddy, D., Sen, A., Johnston, G.: Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Phys. Rev. Lett. 85, 3381–3384 (2000)

    Article  Google Scholar 

  28. Karnatak, R., Ramaswamy, R., Prasad, A.: Amplitude death in the absence of time delays in identical coupled oscillators. Phys. Rev. E 76, 35201 (2007)

    Article  Google Scholar 

  29. Konishi, K.: Amplitude death induced by dynamic coupling. Phys. Rev. E 68, 67202 (2003)

    Article  Google Scholar 

  30. Chen, Y., Xiao, J., Liu, W., Li, L., Yang, Y.: Dynamics of chaotic systems with attractive and repulsive couplings. Phys. Rev. E 80, 46206 (2009)

    Article  Google Scholar 

  31. Prasad, A., Dhamala, M., Adhikari, B.M., Ramaswamy, R.: Amplitude death in nonlinear oscillators with nonlinear coupling. Phys. Rev. E 81, 27201 (2010)

    Article  Google Scholar 

  32. Resmi, V., Ambika, G., Amritkar, R.E.: General mechanism for amplitude death in coupled systems. Phys. Rev. E 84, 46212 (2011)

    Article  Google Scholar 

  33. Sun, X., Xu, J., Fu, J.: The effect and design of time delay in feedback control for a nonlinear isolation system. Mech. Syst. Signal Process. 87, 206–217 (2017)

    Article  Google Scholar 

  34. Ahn, K.H.: Enhanced signal-to-noise ratios in frog hearing can be achieved through amplitude death. J. R. Soc. Interface 10, 20130525 (2013)

    Article  Google Scholar 

  35. Kim, K.J., Ahn, K.H.: Amplitude death of coupled hair bundles with stochastic channel noise. Phys. Rev. E 89, 42703 (2014)

    Article  Google Scholar 

  36. Zhang, H.C., Xu, D.L., Xia, S.Y., Lu, C., Qi, E.R., Tian, C., Wu, Y.S.S., Xia, Y., Lu, C., Qi, E.R., Tian, C., Wu, Y.S.: Nonlinear network modeling of multi-module floating structures with arbitrary flexible connections. J. Fluids Struct. 59, 270–284 (2015)

    Article  Google Scholar 

  37. Zhang, H.C., Xu, D.L., Lu, C., Xia, S.Y., Qi, E.R., Hu, J.J., Wu, Y.S.: Network dynamic stability of floating airport based on amplitude death. Ocean Eng. 104, 129–139 (2015)

    Article  Google Scholar 

  38. Zhang, H.C., Xu, D.L., Lu, C., Qi, E.R., Hu, J.J., Wu, Y.S.: Amplitude death of a multi-module floating airport. Nonlinear Dyn. 79, 2385–2394 (2015)

    Article  Google Scholar 

  39. Xu, D.L., Zhang, H.C., Lu, C., Qi, E.R., Tian, C., Wu, Y.S.: Analytical criterion for amplitude death in nonautonomous systems with piecewise nonlinear coupling. Phys. Rev. E 89, 42906 (2014)

    Article  Google Scholar 

  40. Xiong, Y.P., Xing, J.T., Price, W.G.: A general linear mathematical model of power flow analysis and control for integrated structure-control systems. J. Sound Vib. 267, 301–334 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, H.C., Xu, D.L., Liu, C.R., Wu, Y.S.: Wave energy absorption of a wave farm with an array of buoys and flexible runway. Energy 109, 211–223 (2016)

    Article  Google Scholar 

  42. Watanabe, E., Utsunomiya, T., Wang, C.M.: Hydroelastic analysis of pontoon-type VLFS: a literature survey. Eng. Struct. 26, 245–256 (2004)

    Article  Google Scholar 

  43. Cheng, H.Z., Wang, P., Zou, G.P.: Research on static mechanical properties of metal rubber by wire mesh. Appl. Mech. Mater. 633–634, 238–241 (2014)

    Article  Google Scholar 

  44. Yu, H., Sun, X., Xu, J., Zhang, S.: Transition sets analysis based parametrical design of nonlinear metal rubber isolator. Int. J. Nonlinear Mech. 96, 93–105 (2017)

    Article  Google Scholar 

  45. Stoker, J.J.: Water waves: the mathematical theory with applications. Wiley-Interscience, Hoboken (2011)

    MATH  Google Scholar 

  46. Siddorn, P., Eatock Taylor, R.: Diffraction and independent radiation by an array of floating cylinders. Ocean Eng. 35, 1289–1303 (2008)

    Article  Google Scholar 

  47. Yeung, R.W.: Added mass and damping of a vertical cylinder in finite-depth waters. Appl. Ocean Res. 3, 119–133 (1981)

    Article  Google Scholar 

  48. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1964)

    MATH  Google Scholar 

  49. Balthazar, J.M., Mook, D.T., Weber, H.I., Brasil, M.L.R.F., Fenili, A., Belato, D., Felix, J.L.P.: An overview on non-ideal vibrations. Meccanica 38, 613–621 (2003)

    Article  MATH  Google Scholar 

  50. Park, S., Chung, J.: Dynamic analysis of an axially moving finite-length beam with intermediate spring supports. J. Sound Vib. 333, 6742–6759 (2014)

    Article  Google Scholar 

  51. Felix, J.L.P., Balthazar, J.M., Dantas, M.J.H.: On energy pumping, synchronization and beat phenomenon in a nonideal structure coupled to an essentially nonlinear oscillator. Nonlinear Dyn. 56, 1–11 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Banerjee, T., Biswas, D.: Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion. Chaos 23, 43101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Cao, L.Y., Lai, Y.C.: Antiphase synchronism in chaotic systems. Phys. Rev. E 58, 382–386 (1998)

    Article  Google Scholar 

  54. Zhai, Y., Kiss, I., Hudson, J.: Amplitude death through a Hopf bifurcation in coupled electrochemical oscillators: experiments and simulations. Phys. Rev. E 69, 26208 (2004)

    Article  Google Scholar 

  55. Karnatak, R., Punetha, N., Prasad, A., Ramaswamy, R.: Nature of the phase-flip transition in the synchronized approach to amplitude death. Phys. Rev. E 82, 46219 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

This research work was supported by the National Natural Science Foundation of China (11702088, 11472100), Project funded by China Postdoctoral Science Foundation (2017M620344, 2018T110823) and the High-tech Ship Research Projects Sponsored by MIIT.

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People’s Republic of China

    Haicheng Zhang, Daolin Xu & Shuyan Xia

  2. China Ship Scientific Research Center, Wuxi, 214082, People’s Republic of China

    Yousheng Wu

Authors
  1. Haicheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daolin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuyan Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yousheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daolin Xu.

Appendix

Appendix

The followings are the expressions for the quantities \(\mathbf{q}^{\left( 1 \right) },\mathbf{q}^{\left( 2 \right) }\):

$$\begin{aligned} \mathbf{q}_{1q}^{\left( 1 \right) }= & {} -\frac{1}{\pi }\int _0^{2\pi } {f_{w} (q)\sin \phi \hbox {d}\phi } \\= & {} -\,\frac{1}{\pi }\sum _{i=1}^{N_\mathrm{b} } {\left[ {\varepsilon _1 \varPi _q \left( {X_i } \right) \hbox {IntG1S}} \right] } \\ \mathbf{q}_{2i}^{\left( 1 \right) }= & {} -\,\frac{1}{\pi }\int _0^{2\pi } {\left[ {F_{\mathrm{e}i} \hbox {e}^{\mathrm{i}\omega t}+f_z (i)} \right] \sin \phi \hbox {d}\phi } \\= & {} -\,\frac{1}{\pi }\hbox {IntFS}-\frac{1}{\pi }\left[ \!\! -\varepsilon _1 \hbox {IntG1S}\right. \\&\left. +\varepsilon _2 \sum _{j=1}^{N_\mathrm{b} } {\varPhi _{ij} \hbox {IntG2S}} \right] \\ \mathbf{q}_{1q}^{\left( 2 \right) }= & {} -\,\frac{1}{\pi }\int _0^{2\pi } {f_{w} (q)\cos \phi \hbox {d}\phi } \\= & {} -\frac{1}{\pi }\sum _{i=1}^{N_\mathrm{b} } {\left[ {\varepsilon _1 \varPi _q \left( {X_i } \right) \hbox {IntG1C}} \right] } \\ \mathbf{q}_{2i}^{\left( 1 \right) }= & {} -\frac{1}{\pi }\int _0^{2\pi } {\left[ {F_{\mathrm{e}i} \hbox {e}^{\mathrm{i}\omega t}+f_z (i)} \right] \cos \phi \hbox {d}\phi } \\= & {} -\,\frac{1}{\pi }\hbox {IntFC}-\frac{1}{\pi }\left[ \!\! -\varepsilon _1 \hbox {IntG1C}\right. \\&\quad \left. +\,\varepsilon _2 \sum _{j=1}^{N_\mathrm{b} } {\varPhi _{ij} \hbox {IntG2C}} \right] \end{aligned}$$

where

$$\begin{aligned} \hbox {IntFS}= & {} \int _0^{2\pi } {F_{\mathrm{e}i} \hbox {e}^{\mathrm{i}\omega t}\sin \phi \hbox {d}\phi } \\= & {} \int _0^{2\pi } \left| {F_{\mathrm{e}i} } \right| \left( {\cos \phi \cos \varphi _i +\sin \phi \sin \varphi _i } \right) \\&\sin \phi \hbox {d}\phi =\pi \left| {F_{\mathrm{e}i} } \right| \sin \varphi _i \end{aligned}$$
$$\begin{aligned} \hbox {IntFC}= & {} \int _0^{2\pi } {F_{\mathrm{e}i} \hbox {e}^{\mathrm{i}\omega t}\cos \phi \hbox {d}\phi } \\= & {} \int _0^{2\pi } \left| {F_{\mathrm{e}i} } \right| \left( {\cos \phi \cos \varphi _i +\sin \phi \sin \varphi _i } \right) \\&\cos \phi \hbox {d}\phi =\pi \left| {F_{\mathrm{e}i} } \right| \cos \varphi _i \end{aligned}$$
$$\begin{aligned} \hbox {IntG1S}= & {} \int _0^{2\pi } {G_1 \left( {z_i ,w_j } \right) \sin \left( \phi \right) \hbox {d }\phi }\\= & {} \frac{3}{4}\pi \left[ \left( {u_{2i} -\sum _{j=0}^{N_\mathrm{m} } {u_{1j} \varPi _j (X_i )} } \right) ^{2}\right. \\&\left. +\,\left( {v_{2i} -\sum _{j=0}^{N_\mathrm{m} } {v_{1j} \varPi _j (X_i )} } \right) ^{2} \right] \\&\left( {v_{2i} -\sum _{j=0}^{N_\mathrm{m} } {v_{1j} \varPi _j (X_i )} } \right) \\ \hbox {IntG1C}= & {} \int _0^{2\pi } {G_1 \left( {z_i ,w_j } \right) \cos \left( \phi \right) \hbox {d }\phi }\\= & {} \frac{3}{4}\pi \left[ \left( {u_{2i} -\sum _{j=0}^{N_\mathrm{m} } {u_{1j} \varPi _j (X_i )} } \right) ^{2}\right. \\&\left. +\,\left( {v_{2i} -\sum _{j=0}^{N_\mathrm{m} } {v_{1j} \varPi _j (X_i )} } \right) ^{2} \right] \\&\left( {u_{2i} -\sum _{j=0}^{N_\mathrm{m} } {u_{1j} \varPi _j (X_i )} } \right) \\ \hbox {IntG2S}= & {} \int _0^{2\pi } {G_2 \left( {z_i ,z_j } \right) \sin \left( \phi \right) \hbox {d }\phi } \\= & {} \pi \left( {v_{2j} -v_{2i} } \right) \\ \hbox {IntG2C}= & {} \int _0^{2\pi } {G_2 \left( {z_i ,z_j } \right) \cos \left( \phi \right) \hbox {d }\phi } \\= & {} \pi \left( {u_{2j} -u_{2i} } \right) \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Xu, D., Xia, S. et al. Nonlinear dynamics of a non-autonomous network with coupled discrete–continuum oscillators. Nonlinear Dyn 94, 889–904 (2018). https://doi.org/10.1007/s11071-018-4400-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4400-1

Keywords

Search

Navigation