Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Breakup of Closed Curve - Quasiperiodic Route to Chaos in Vibroimpact System

Discontinuity, Nonlinearity, and Complexity 8(3) (2019) 299--311 | DOI:10.5890/DNC.2019.09.006

V. A. Bazhenov, O. S. Pogorelova, T. G. Postnikova

Kyiv National University of Construction and Architecture, 31, Povitroflotskiy avenu, Kyiv, Ukraine

Download Full Text PDF



At present chaotic vibrations are the one of the most interesting and explored subjects in nonlinear dynamics. Particularly the routes to chaos in non-smooth dynamical systems are of the special scientists’ interest. Quasiperiodic route to chaos in nonlinear non-smooth discontinuous 2-DOF vibroimpact system is studied in this paper. In narrowfrequency range different oscillatory regimes have succeeded each other many times under very small control parameter varying. There were periodic subharmonic regimes - chatters, quasiperiodic, and chaotic regimes. There were the zones of transition from one regime to another, the zones of prechaotic and postchaotic motion. The hysteresis effects (jump phenomena) occurred for increasing and decreasing frequencies. The chaoticity of obtained regime has been confirmed by typical views of Poincaré map and Fourier spectrum, by the positive value of the largest Lyapunov exponent, and by the fractal structure of Poincaré map.


  1. [1]  Bernardo, M., Budd, C., Champneys, A. R., and Kowalczyk, P. (2008), Piecewise-smooth dynamical systems: theory and applications, 163, Springer Science & Business Media.
  2. [2]  Luo, A.C. and Guo, Y. (2012), Vibro-impact dynamics, John Wiley & Sons.
  3. [3]  Leine, R.I. and Van Campen, D.H. (2002), Discontinuous bifurcations of periodic solutions, Mathematical and computer modelling, 36(3), 259-273.
  4. [4]  Seydel, R. (2009), Practical bifurcation and stability analysis, 5, Springer Science & Business Media.
  5. [5]  Ivanov, A.P. (2012), Analysis of discontinuous bifurcations in nonsmooth dynamical systems, Regular and Chaotic Dynamics, 17(3-4), 293-306.
  6. [6]  Leine, R.I., Van Campen, D.H., and Van de Vrande, B.L. (2000), Bifurcations in nonlinear discontinuous systems, Nonlinear dynamics, 23(2), 105-164.
  7. [7]  Kowalczyk, P., di Bernardo, M., Champneys, A.R., Hogan, S.J., Homer, M., Piiroinen, P.T., and Nordmark, A. (2006), Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems, International Journal of Bifurcation and Chaos, 16(03), 601-629.
  8. [8]  Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A B., Tost, G.O., and Piiroinen, P.T. (2008), Bifurcations in nonsmooth dynamical systems, SIAM review, 50(4), 629-701.
  9. [9]  Brogliato, B. (1999), Nonsmooth mechanics, London: Springer-Verlag.
  10. [10]  Brogliato, B. (Ed.). (2000), Impacts in mechanical systems: analysis and modelling, 551, Springer Science & Business Media.
  11. [11]  Alzate, R. (2008), Analysis and Application of Bifurcations in Systems with Impacts and Chattering (Doctoral dissertation, Universit`a degli Studi di Napoli Federico II).
  12. [12]  di Bernardo, M., Budd, C. J., Champneys, A.R., and Kowalczyk (2007), Bifurcation and Chaos in Piecewise Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, UK.
  13. [13]  Cheng, J. and Xu, H. (2006), Nonlinear dynamic characteristics of a vibro-impact system under harmonic excitation, Journal of Mechanics of Materials and Structures, 1(2), 239-258.
  14. [14]  Volchenkov, D. (2018), Success, Hierarchy, and Inequality Under Uncertainty, In Regularity and Stochasticity of Nonlinear Dynamical Systems (pp. 51-78), Springer, Cham.
  15. [15]  Shil'nikov, L.P. (2001), Methods of qualitative theory in nonlinear dynamics (Vol. 5), World Scientific.
  16. [16]  Thompson, J.M T. and Stewart, H.B. (2002), Nonlinear dynamics and chaos, John Wiley & Sons.
  17. [17]  Kuznetsov, S.P. (2001), Dynamical chaos. Moscow: Fizmatlit.-2006.-356P.
  18. [18]  Moon, F.C. (1987), Chaotic vibrations: an introduction for applied scientists and engineers, Research supported by NSF, USAF, US Navy, US Army, and IBM. New York,Wiley-Interscience, 1987, 322 p.
  19. [19]  Luo, A.C. (2016), Periodic Flows to Chaos in Time-delay Systems, 16, Springer.
  20. [20]  Afraimovich, V.S. and Neiman, A.B. (2017), Weak transient chaos, In Advances in Dynamics, Patterns, Cognition, Springer, Cham., 3-12.
  21. [21]  Afraimovich, V., Machado, J A.T., and Zhang, J. (Eds.). (2016), Complex Motions and Chaos in Nonlinear Systems, Springer International Publishing.
  22. [22]  Shvets, A.Y. and Sirenko, V.A. (2015), New Ways of Transition to Deterministic Chaos in Nonideal Oscillating Systems, Research Bulletin of National Technical University of Ukraine "Kyiv Polytechnic Institute", 1(99), 45-51.
  23. [23]  Serweta, W., Okolewski, A., Blazejczyk-Okolewska, B., Czolczynski, K., and Kapitaniak, T. (2014), Lyapunov exponents of impact oscillators with Hertz's and Newton's contact models, International Journal of Mechanical Sciences, 89, 194-206.
  24. [24]  Manneville, P. and Pomeau, Y. (1980), Different ways to turbulence in dissipative dynamical systems, Physica D: Nonlinear Phenomena, 1(2), 219-226.
  25. [25]  Müller, P.C. (1995), Calculation of Lyapunov exponents for dynamic systems with discontinuities, Chaos, Solitons & Fractals, 5(9), 1671-1681.
  26. [26]  Stefanski, A. (2000), Estimation of the largest Lyapunov exponent in systems with impacts, Chaos, Solitons& Fractals, 11(15), 2443-2451.
  27. [27]  Stefanski, A. (2000), Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems, Discrete Dynamics in Nature and society, 4(3), 207-215.
  28. [28]  Stefański, A. and Kapitaniak, T. (2003), Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization, Chaos, Solitons & Fractals, 15(2), 233-244.
  29. [29]  De Souza, S.L. and Caldas, I.L. (2004), Calculation of Lyapunov exponents in systems with impacts, Chaos, Solitons & Fractals, 19(3), 569-579.
  30. [30]  Stefanski, A., Dabrowski, A., and Kapitaniak, T. (2005), Evaluation of the largest Lyapunov exponent in dynamical systems with time delay, Chaos, Solitons & Fractals, 23(5), 1651-1659.
  31. [31]  Ageno, A. and Sinopoli, A. (2005), Lyapunov's exponents for nonsmooth dynamics with impacts: Stability analysis of the rocking block, International Journal of Bifurcation and Chaos, 15(06), 2015-2039.
  32. [32]  Andreaus, U., Placidi, L., and Rega, G. (2010), Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator, Communications in Nonlinear Science and Numerical Simulation, 15(9), 2603-2616.
  33. [33]  Li, Q.H. and Tan, J.Y. (2011), Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops, Chinese Physics B, 20(4), 040505.
  34. [34]  Li, Q., Chen, Y., Wei, Y., and Lu, Q. (2011), The analysis of the spectrum of Lyapunov exponents in a two-degree-offreedom vibro-impact system, International Journal of Non-Linear Mechanics, 46(1), 197-203.
  35. [35]  Baumann, M. (2017), Synchronization of nonsmoothmechanical systems with impulsive motion (Doctoral dissertation, ETH Zurich).
  36. [36]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2017), Stability and Discontinious Bifurcations in Vibroimpact System: Numerical investigations, LAP LAMBERT Academic Publ. GmbH and Co. KG Dudweiler, Germany.
  37. [37]  Bazhenov, V.A., Lizunov, P.P., Pogorelova, O.S., Postnikova, T.G., and Otrashevskaia, V.V. (2015), Stability and Bifurcations Analysis for 2-DOF Vibroimpact System by Parameter Continuation Method. Part I: Loading Curve, Journal of Applied Nonlinear Dynamics, 4(4), 357-370, DOI: 10.5890/JAND.2015.11.003.
  38. [38]  Bazhenov, V.A., Lizunov, P.P., Pogorelova, O.S., and Postnikova, T.G. (2016), Numerical Bifurcation Analysis of Discontinuous 2-DOF Vibroimpact System, Part 2: Frequency-Amplitude response, Journal of Applied Nonlinear Dynamics, 5(3), 269鈥?81. DOI: 10.5890/JAND.2016.09.002
  39. [39]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2016), Dangerous bifurcations in 2-dof vibroimpact system, The Bulletin of the National Technical University "Kharkiv Polytechnic Institute", (26), 109-113.
  40. [40]  Lamarque, C.H. and Janin, O. (2000), Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition, Journal of Sound and Vibration, 235(4), 567-609.
  41. [41]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2018), Lyapunov exponents estimation for strongly nonlinear nonsmooth discontinuous vibroimpact system, Strength of Materials and Theory of Structures, (99), (in press).