Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


On Experimental Periodic Motions in a Duffing Oscillatory Circuit

Journal of Vcibration Testing and System Dynamics 3(1) (2019) 55--69 | DOI:10.5890/JVTSD.2019.03.005

Yu Guo$^{1}$, Albert C.J. Luo$^{2}$, Zeltzin Reyes$^{1}$, Abigail Reyes$^{1}$, Rojitha Goonesekere$^{1}$

$^{1}$ McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA

$^{2}$ Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA

Download Full Text PDF



In this paper, the dynamics of a Duffing oscillatory system are studied through both an experimental method and analytical predictions. A Duffing oscillatory circuit is built and experimental data of various periodic motions are obtained. The semi-analytical method using discrete implicit maps is adopted to determine such periodic trajectories analytically using the same parameters as in the experiments. Finally, illustrations of various periodic motions on both trajectories and harmonic amplitudes are demonstrated between the experimental data and analytical predictions. From the current experimental technology, experimental results may not provide real results in nonlinear dynamical systems.


  1. [1]  Lagrange, J.L. (1788), Mecanique Analytique, (2 vol.) (edition Albert Balnchard: Paris, 1965).
  2. [2]  Poincare, H. (1899), Methodes Nouvelles de la Mecanique Celeste, Vol.3, Gauthier-Villars: Paris.
  3. [3]  van der Pol, B. (1920), A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, pp.701-710, pp.754-762.
  4. [4]  Fatou, P. (1928), Sur le mouvement d’un systeme soumis ‘a des forces a courte periode, Bull. Soc. Math., 56, 98-139.
  5. [5]  Krylov, N.M. and Bogolyubov, N.N. (1935), Methodes approchees de la mecanique non-lineaire dans leurs application a l’Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s’y rapportant, Academie des Sciences d’Ukraine:Kiev. (in French).
  6. [6]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon.
  7. [7]  Luo, A.C.J. and Huang, J.Z. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance, Journal of Vibration and Control, 18, 1661-1871.
  8. [8]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093 (29 pages).
  9. [9]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential, Journal of Applied Nonlinear Dynamics, 1, 73-108.
  10. [10]  Luo, A.C.J. and Huang, J.Z. (2012), Unstable and stable period-m motions in a twin-well potential Duffing oscillator, Discontinuity, Nonlinearity and Complexity, 1, 113-145.
  11. [11]  Luo, A.C.J. (2005), The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, Journal of Sound and Vibration, 283, 723-748.
  12. [12]  Luo, A.C.J. (2005), A theory for non-smooth dynamic systems on the connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10, 1-55.
  13. [13]  Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer: New York.
  14. [14]  Luo, A.C.J. (2015), Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems, International Journal of Bifurcation and Chaos, 25(3), Article No. 1550044 (62 pages).
  15. [15]  Luo, A.C.J. and Guo, Y. (2015), A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures, Discontinuity, Nonlinearity, and Complexity, 4(2), pp.13-44.
  16. [16]  Guo, Y. and Luo, A.C.J. (2015), Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings, International Journal of Dynamics and Control, 5(2), 223-238.
  17. [17]  Luo, A.C.J. and Guo, Y. (2016), Periodic motions to chaos in pendulum, International Journal of Bifurcation and Chaos, 26(9), 1650159.
  18. [18]  Guo, Y. and Luo, A.C.J. (2017), Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3), 551-569.
  19. [19]  Chua, L.O., Wu, C.W., Huang, A., and Zhong, G.Q. (1993), A universal circuit for studying and generating chaos, Part I, II, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40(10), 732-761.
  20. [20]  Kapitaniak T., Chua, L.O., and Zhong, G.Q. (1997), Experimental evidence of locally intermingled basins of attraction in coupled Chua’s circuits, Chaos, Solitons & Fractals, 8(9), 1517-1522
  21. [21]  Tafo Wembe, E. and Yamapi, R. (2009), Chaos synchronization of resistively coupled Duffing systems: numerical and experimental investigations, Communications in Nonlinear Science and Numerical Simulation, 14(2009), 1439-1453.
  22. [22]  Trejo-Guerra, R., Tlelo-Cuautle, E., Carbajal-Gomez, V.H., and Rodriguez-Gomez, G. (2013), A survey on the integrated design of chaotic oscillators, Applied Mathematics and Computation, 219, 5113-5122.
  23. [23]  Buscarino, A., Fortuna, L., Frasca, M., and Scuito, G. (2014), A concise guide to chaotic electronic circuits, Springer Briefs in Applied Sciences and Technology.