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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


Unstable and Stable Period-m Motions in a Twin-well Potential Duffing Oscillator

Discontinuity, Nonlinearity, and Complexity 1(2) (2012) 113--145 | DOI:10.5890/DNC.2012.04.001

Albert C. J. Luo; Jianzhe Huang

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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In this paper, unstable and stable period-m motions in the periodically forced Duffing oscillator are predicted analytically through the generalized harmonic balance method. Period-3, period-5 and period-7 periodic motions are investigated as examples for the Duffing oscillator with a twin-well potentials. The Hopf bifurcation of periodic motions yields the onset of period-doubling periodic motions. With increasing period number of periodic motions, there are too many co-existing stable and unstable periodic motions, and such stable periodic motions are much less than the corresponding unstable periodic motions. This investigation provides a complete picture of unstable and stable periodic motions rather than stable motions only. For any unstable periodic motion, if there is at least one co-existing stable periodic motion, then such unstable periodic motion will reach the stable periodic motion through transient motion.


  1. [1]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical routine to period-1 motions to chaos in a periodically forced Duffing oscillator with a twin-well potential, Journal of Applied Nonlinear Dynamics, 1, 73-108
  2. [2]  Duffing, G. (1918), Erzwunge Schweingungen bei veranderlicher eigenfrequenz. Braunschweig: F. Viewig u. Sohn.
  3. [3]  Hayashi, G. (1964), Nonlinear Oscillations in Physical Systems, McGraw-Hill Book Company: New York
  4. [4]  Nayfeh, A.H. (1973), Perturbation Methods, JohnWiley: New York.
  5. [5]  Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillation, JohnWiley: New York.
  6. [6]  Holmes, P.J. (1979), A nonlinear oscillator with strange attractor, Philosophical Transactions of the Royal Society A, 292, 419-448.
  7. [7]  Ueda, Y. (1980), Explosion of strange attractors exhibited by the Duffing equations, Annuals of the New York Academy of Science, 357, 422-434.
  8. [8]  Luo, A.C.J. and Han, R.P.S. (1997), A quantitative stability and bifurcation analyses of a generalized Duffing oscillator with strong nonlinearity, Journal of Franklin Institute, 334B, 447-459.
  9. [9]  Han, R.P.S. and Luo, A.C.J. (1996), Comments on "Subharmonic resonances and criteria for escape and chaos in a driven oscillator", Journal of Sound and Vibration, 196(2), 237-242.
  10. [10]  Luo, A.C.J. and Han, R.P.S. (1999), Analytical predictions of chaos in a nonlinear rod, Journal of Sound and Vibration, 227(2), 523-544.
  11. [11]  Luo, A.C.J. (2008), Global Transversality, Resonance and Chaotic Dynamics,World Scientific, New Jersey.
  12. [12]  Garcia-Margallo and J. & Bejarano, J. D. (1987), A generalization of the method of harmonic balance, Journal of Sound and Vibration, 116, 591-595.
  13. [13]  Rand, R.H. andArmbruster, D. (1987), PerturbationMethods, Bifurcation Theory, and Computer Algebra.Applied Mathematical Sciences, no. 65, Springer-Verlag: New York.
  14. [14]  Yuste, S.B. and Bejarano, J.D. (1989), Extension and improvement to the Krylov-Bogoliubov method that use elliptic functions, International Journal of Control, 49, 1127-1141.
  15. [15]  Coppola, V.T. and Rand, R.H. (1990), Averaging using elliptic functions: Approximation of limit cycle, Acta Mechanica, 81, 125-142.
  16. [16]  Peng, Z. K., Lang, Z. Q., Billings, S. A. and Tomlinson, G. R. (2008), Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis, Journal of Sound and Vibration, 311, 56-73.
  17. [17]  Luo, A.C.J. and Huang, J.Z. (2011), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance, Journal of Vibration and Control, in press.
  18. [18]  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, 1250093 (29 pages).
  19. [19]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP-L&HScientific, Glen Carbon.