ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

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

A Semi-analytical Prediction of Periodic Motions in Duffing Oscillator Through Mapping Structures

Discontinuity, Nonlinearity, and Complexity 4(2) (2016) 121--150 | DOI:10.5890/DNC.2016.06.002

Albert C.J. Luo; Yu Guo

$^{1}$ Department ofMechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

$^{2}$ McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA

Abstract

In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos.

References

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, 701-710, 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] Hayashi, C. (1964), Nonlinear oscillations in Physical Systems, McGraw-Hill Book Company, New York.
7.  [7] Barkham, P.G.D. and Soudack, A.C. (1969), An extension to the method of Krylov and Bogoliubov, International Journal of Control, 10, 377-392.
8.  [8] Barkham, P.G.D. and Soudack, A.C. (1970), Approximate solutions of nonlinear, non-autonomous second-order differential equations, International Journal of Control, 11, 763-767.
9.  [9] Rand, R.H. and Armbruster, D. (1987), Perturbation Methods, Bifurcation Theory, and Computer Algebra, Applied Mathematical Sciences, 65, Springer-Verlag, New York.
10.  [10] Garcia-Margallo, J. and Bejarano, J.D. (1987), A generalization of the method of harmonic balance, Journal of Sound and Vibration, 116, 591-595.
11.  [11] 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.
12.  [12] Coppola, V.T. and Rand, R.H. (1990), Averaging using elliptic functions: Approximation of limit cycle, Acta Mechanica, 81, 125-142.
13.  [13] Luo, A.C.J. (2012) Continuous Dynamical Systems, HEP/L&H Scientific, Beijing/Glen Carbon.
14.  [14] 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.
15.  [15] 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).
16.  [16] 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.
17.  [17] 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.
18.  [18] 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.
19.  [19] 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.
20.  [20] Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer, New York.
21.  [21] Luo, A.C.J. (2009), Discontinuous Dynamical Systems on Time-varying Domains, Higher Education Press/Springer, Beijing/Heidelberg.
22.  [22] Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25(3), Article No. 1550044 (62 pages).