Period-m Motions to Chaos in a Periodically Forced, Duffing Oscillator with a Time-Delayed Displacement

In this paper, periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The time-delayed displacement is from the feedback control of displacement. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator are presented through asymmetric period-1 to period-4 motions. Stable and unstable periodic motions are illustrated through numerical and analytical solutions. From numerical illustrations, the analytical solutions of stable and unstable period-m motions are relatively accurate with AN/m < 10-6 compared to numerical solutions. From such analytical solutions, any complicated solutions of period-m motions can be obtained for any prescribed accuracy. Because time-delay may cause discontinuity, the appropriate time-delay inputs (or initial conditions) in the initial time-delay interval should satisfy the analytical solution of periodic motions in the time-delayed dynamical systems. Otherwise, periodic motions in such a time-delayed system cannot be obtained directly.

[1]  A. Luo.  Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay , 2013 .

[2]  Jiří Tlustý,et al.  Manufacturing processes and equipment , 1999 .

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

[4]  Vimal Singh,et al.  Perturbation methods , 1991 .

[5]  N. Bogolyubov,et al.  Asymptotic Methods in the Theory of Nonlinear Oscillations , 1961 .

[6]  malaman.  Mécanique analytique I , 2016 .

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

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

[9]  N. Macdonald,et al.  Harmonic balance in delay-differential equations , 1995 .

[10]  Tamás Kalmár-Nagy,et al.  High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations , 2010 .

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

[12]  Albert C. J. Luo,et al.  Unstable and Stable Period-m Motions in a Twin-well Potential Duffing Oscillator , 2012 .

[13]  Earl H. Dowell,et al.  Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback , 1998 .

[14]  H. Hu,et al.  Dynamics of Controlled Mechanical Systems with Delayed Feedback , 2002 .

[15]  Haiyan Hu,et al.  Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations , 2003 .

[16]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[17]  Albert C. J. Luo,et al.  Analytical Routes of Period-1 Motions to Chaos in a Periodically Forced Duffing Oscillator with a Twin-well Potential , 2012 .

[18]  Andrew Y. T. Leung,et al.  Bifurcation of the periodic motion in nonlinear delayed oscillators , 2014 .

[19]  A. Luo,et al.  Bifurcation Trees of Period-m Motions to Chaos in a Time-Delayed, Quadratic Nonlinear Oscillator under a Periodic Excitation , 2014 .

[20]  Jian-Qiao Sun,et al.  A method of continuous time approximation of delayed dynamical systems , 2009 .

[21]  A. Luo,et al.  Complex period-1 motions of a periodically forced Duffing oscillator with a time-delay feedback , 2015 .

[22]  Gábor Stépán,et al.  Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications , 2011 .