Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator

In this paper, analytical solutions for complex period-1 motions in a periodically forced, quadratic nonlinear oscillator are presented through the Fourier series solutions with finite harmonic terms, and the corresponding stability and bifurcation analyses of the corresponding period-1 motions are carried out. Many branches of complex period-1 motions in such a quadratic nonlinear oscillator are discovered and the period-1 motion patterns changes with parameters are presented. The parameter map for excitation amplitude and frequency is developed for different complex period-1 motions. For small excitation frequency, the period-1 motion becomes more complicated. For a better understanding of complex period-1 motions in such a quadratic nonlinear oscillator, trajectories and amplitude spectrums are illustrated numerically. From stability and bifurcations analysis of the period-1 motion, the analytical bifurcation trees of period-1 motions to chaos need to be further investigated.

[1]  Ronald E. Mickens,et al.  A generalization of the method of harmonic balance , 1986 .

[2]  P. Fatou,et al.  Sur le mouvement d'un système soumis à des forces à courte période , 1928 .

[3]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[4]  Albert C. J. Luo,et al.  Analytical Dynamics of Period-M Flows and Chaos in nonlinear Systems , 2012, Int. J. Bifurc. Chaos.

[5]  E. Thandapani,et al.  Approximate solutions of nonlinear differential equations , 1988 .

[6]  J. Garcia-Margallo,et al.  A generalization of the method of harmonic balance , 1987 .

[7]  S. Bravo Yuste,et al.  Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions , 1989 .

[8]  Albert C. J. Luo,et al.  A quantitative stability and bifurcation analyses of the generalized duffing oscillator with strong nonlinearity , 1997 .

[9]  G. Adomian,et al.  Nonlinear oscillations in physical systems , 1987 .

[10]  Approximate solutions of non-linear non-autonomous second-order differential equations† , 1970 .

[11]  Albert C. J. Luo,et al.  Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance , 2012 .

[12]  Albert C. J. Luo,et al.  Analytical Solutions for Stable and Unstable Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity , 2013 .

[13]  A. C. Soudack,et al.  An extension to the method of Kryloff and Bogoliuboff , 1969 .

[14]  Albert C. J. Luo.  Continuous Dynamical Systems , 2012 .

[15]  F. Cap,et al.  Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions , 1974 .

[16]  Vincent T. Coppola,et al.  Averaging using elliptic functions: approximation of limit cycles , 1990 .

[17]  G. Tomlinson,et al.  Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis , 2008 .