Analytical solutions and bifurcation of nonlinear oscillators with discontinuities and impulsive systems by a perturbation-incremental method


Student thesis: Doctoral Thesis

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  • Hailing WANG

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Awarding Institution
Award date3 Oct 2012


Nonlinear equations have been widely used in many areas of physics and engineering. They are of significant importance in mechanical and structural dynamics for the comprehensive understanding and accurate prediction of motion. Analytical solution obtained from classical perturbation methods such as Lindstedt-Poincar'e method, Krylov-Bogoliubov-Mitropolsky method, method of multiple scales and averaging method are usually accurate for small perturbation. For nonlinear oscillators with discontinuities, accurate analytical solution may not be easily obtained due to the nonsmooth property at the switching points. For impulsive systems, there is a sudden jump in the phase portrait. To the best of our knowledge, no harmonic balance method has ever been applied to investigate the bifurcation and continuation of period solutions of such systems. In this thesis, we investigate analytical solutions of nonlinear oscillators with discontinuities using a nonlinear time transformation method, and bifurcation and continuation of impulsive systems using a perturbation-incremental method. First, we study analytical periodic solutions of a generalized Duffing-harmonic oscillator having a rational form for the potential energy by a nonlinear time transformation method. An analytical solution is expressed in Pad'e approximation which often gives a better approximation of a function than its truncating Taylor series. Period solutions with large amplitude and those near to homoclinic/heteroclinic orbits are computed. Excellent agreement of the approximate presentations with the numerical simulation has been demonstrated and discussed. We also compared the results with those from the cubication method. Next, we present a nonlinear time transformation method to obtain analytical solutions of nonlinear oscillators with discontinuities. The essence of this method is that a periodic solution is approximated by the Chebyshev polynomials with a nonlinear time s rather than the physical time t. Since the first derivative of an approximate limit cycle oscillation obtained from the present method is piecewise continuous which agrees qualitatively with the exact solution, it gives accurate analytical solutions for the nonlinear oscillators with discontinuities. In some cases, the present method gives exact solutions while other perturbation methods give only approximate solutions. For those systems where exact solution is impossible, the approximate solution obtained from the present method is compared to He's homotopy perturbation method which is a powerful method with good accuracy for many systems. Finally, a perturbation-incremental (PI) method is presented for the bifurcation analysis of periodic solutions of impulsive systems. For such systems, a periodic solution is also approximated by the Chebyshev polynomials instead of the Fourier series so as to overcome the sudden jump in the phase portrait. In the perturbation step, a perturbed solution is obtained at bifurcation through solving a system of low-dimensional linear equations and is taken as an initial guess for incremental iteration. Through an incremental process, period solutions can be calculated to any desired degree of accuracy and their stabilities can be determined by the Floquet theory. As the parameter varies, period-doubling solutions leading to chaos can be identified.

    Research areas

  • Mathematical models, Nonlinear oscillations, Bifurcation theory