Abstract
In this paper, bifurcation trees of periodic motions to chaos in a damped, parametric Duffing oscillator are investigated. From the semi-analytic method, differential equations of nonlinear dynamical systems are discretized first to obtain implicit mappings. Following the implicit mapping structures, periodic nodes of periodic motions are computed. The bifurcation trees of period-1 to period-4 motions are presented to demonstrate the routes of period-1 motions to chaos, and the corresponding stability and bifurcation are determined by eigenvalue analysis. For a better understanding of nonlinear behaviors of periodic motions in a parametric Duffing oscillator, harmonic frequency–amplitude characteristics of periodic motions are presented. From the analytical predictions, numerical simulations are performed. The trajectory, time-histories of displacement and velocity, harmonic amplitudes and phases of period-1 to period-4 motions are presented. Based on comparison of numerical and analytical results, determined is how many harmonic terms should be included in finite Fourier series, which help one select harmonic terms in analytical solutions and engineering application.
Similar content being viewed by others
References
Lagrange JL (1788) Mecanique Analytique, vol 2. Albert Balnchard, Paris, 1965
Poincare H (1899) Methodes Nouvelles de la Mecanique Celeste, vol 3. Gauthier-Villars, Paris
van der Pol B (1920) A theory of the amplitude of free and forced triode vibrations. Radio Rev 1(701–710):754–762
Krylov NM, Bogolyubov NN (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)
Hayashi C (1964) Nonlinear oscillations in physical systems. McGraw-Hill Book Company, New York
Nayfeh AH (1973) Perturbation methods. Wiley, New York
Holmes PJ, Rand DA (1976) Bifurcations of Duffing’s equation; an application of catastrophe theory. Q Appl Math 44:237–253
Nayfeh AH, Mook DT (1979) Nonlinear oscillation. Wiley, New York
Ueda Y (1980) Explosion of strange attractors exhibited by the Duffing equations. Ann N Y Acad Sci 357:422–434
Garica-Margallo JD, Bejarano JD (1987) A generalization of the method of harmonic balance. J Sound Vib 116:591–595
Coppola VT, Rand RH (1990) Averaging using elliptic functions: approximation of limit cycle. Acta Mech 81:125–142
Mathieu E (1873) Memoire sur le mouvement vibra-toire d’une membrance deforme elliptique. Journal de mathematiques pures et appliquees 2 serie, tome 13:137–203 (in French)
Whittaker ET (1913) General solution of Mathieu’s equation. Proc Edinb Amath Soc 32:75–80
Sevin E (1961) On the parametric excitation of pendulum-type vibration absorber. J Appl Mech 28:330–334
Tso WK, Caughey TK (1965) Parametric excitation of a nonlinear system. ASME J Appl Mech 32:899–902
Mond M, Cederbaum G, Khan PB, Zarmi Y (1993) Stability analysis of non-linear Mathieu equation. J Sound Vib 167:77–89
Luo ACJ, Han RPS (1997) A quantitative stability and bifurcation analysis of a generalized Duffing oscillator with strong nonlinearity. J Sound Vib 334B:447–459
Luo ACJ, Han RPS (1999) Analytical prediction of chaos in a nonlinear rod. J Sound Vib 227(3):523–544
Luo ACJ (2004) Chaotic notion in the resonant separatrix bands of a Mathieu–Duffing oscillator with twin-well potential. J Sound Vib 273:653–666
Peng ZK, Lang ZQ, Billings SA, Tomlinson GR (2008) Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis. J Sound Vib 311:56–73
Luo ACJ, Huang JZ (2012) Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. J Sound Vib 18:1661–1871
Luo ACJ (2012) Continuous dynamical systems. HEP/L&H Scientific, Beijing/Glen Carbon
Luo ACJ, Yu B (2013) Period-m motions and bifurcation trees in a periodically excited, quadratic nonlinear oscillator. Discontin Nonlinearity Complex 2:263–288
Luo ACJ, Laken AB (2013) Analytical solutions for period-m motions in a periodically forced van der Pol oscillator. In J Dyn Control 1(2):99–115
Luo ACJ, O’Connor D (2013) On periodic motions in a parametric hardening Duffing oscillator. Int J Bifur Chaos 24:No. 1430004
Luo ACJ (2015) Periodic flows to Chaos based on discrete implicit mappings of continuous nonlinear systems. Int J Bifur Chaos 25(3):No. 1550044
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Luo, A.C.J., Ma, H. Bifurcation trees of periodic motions to chaos in a parametric Duffing oscillator. Int. J. Dynam. Control 6, 425–458 (2018). https://doi.org/10.1007/s40435-017-0314-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-017-0314-x