Period-1 Motions in a Quadratic Nonlinear Oscillator
Analytical solutions for period-1 motions in a periodically forced, quadratic nonlinear oscillator are presented through the Fourier series solutions with finite harmonic terms, and the stability and bifurcation analyses of the corresponding period-1 moti
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Abstract Analytical solutions for period-1 motions in a periodically forced, quadratic nonlinear oscillator are presented through the Fourier series solutions with finite harmonic terms, and the stability and bifurcation analyses of the corresponding period-1 motions are carried out. The parameter map for excitation amplitude and frequency is developed for different period-1 motions. For a better understanding of complex period-1 motions in such a quadratic nonlinear oscillator, trajectories and amplitude spectrums are illustrated numerically.
1 Introduction To obtain analytical solutions of nonlinear dynamical systems is an important issue for a better understanding of nonlinear behaviors and multiplicity. So far, one cannot find an efficient way to determine complex periodic motions in nonlinear dynamical systems yet. In the nineteenth century, Poincare [1] further developed the perturbation theory for celestial bodies. In 1920, van der Pol [2] used the method of averaging to determine the periodic solutions of oscillation systems in circuits. In 1964, Hayashi [3] used perturbation methods, averaging method, and principle of harmonic balance to determine the approximate solutions of nonlinear oscillators. In 1973, Nayfeh [4] presented the multi-scale perturbation method and applied such a perturbation method for obtaining approximate solutions of periodic motions in nonlinear oscillations of structures (also see, Nayfeh and Mook [5]). In 1997, Luo and Han [6] studied the stability and bifurcations of periodic solutions of Duffing oscillators through the first order harmonic balance method. In 2008, Peng et al. [7] presented the approximate period-1 solution for the Duffing oscillator by the HB3
A.C.J. Luo () • B. Yu Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA e-mail: [email protected]; [email protected] V. Afraimovich et al. (eds.), Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 8, DOI 10.1007/978-3-319-02353-3__4, © Springer International Publishing Switzerland 2014
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method and compared with the fourth-order Runge–Kutta method. In 2011, Luo and Huang [8] discussed approximate solutions of periodic motions in nonlinear systems through the harmonic balance method. In 2012, Luo and Huang [9] developed the approximate analytical solutions of period-m motions and chaos. In Luo [10], the methodology and procedure for analytical solutions of periodic motions in general nonlinear systems were presented. Such a method will be used to develop analytical solutions of periodic motions in a quadratic nonlinear system under a periodic excitation. Such an oscillator can be used to describe ship motion under periodic ocean waves. The analytical solutions will include enough harmonic terms to give an appropriate solution of period-1 motion from which the analytical bifurcations and period-m motions can be achieved.
2 Analytical Solutions As in Luo and Yu [11, 12], consider a periodically forced, nonlinear oscillator as xR C ı xP C ˛x C ˇx 2 D Q0 cos
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