Asymptotic Solution Of Nonlinear Oscillator And Numerical Verification

The modified Lindstedt-Poincarémethod modifies the expansion of the fundamental frequency based on the classical L-P method, and the convolution integral method provides an iteration scheme to obtain the asymptotic solution. We obtain the second order solutions of a quadratic nonlinear oscillator respectively by these two methods, and demonstrate that the solution obtained by convolution integral is uniformly convergent. A technique of numerical order verification is applied to verify that the asymptotic solutions are uniformly valid when the parameter is small. But the asymptotic solutions are invalid due to their increasing error when the parameter becomes larger. The reason is that the expansions of frequency obtained by the two methods are valid only for small values of parameter. So, these two methods are limited by small parameter when they are applied to quadratic nonlinear oscillator.A generalized Van del Pol oscillator with slowly varying parameter from a modified Van del Pol oscillator is studied. We obtain three approximate cubic strongly nonlinear oscillators respectively by Taylor series expansions method, approximate potential method and equivalent nonlinearization method. The leading order approximate solutions of these three approximate cubic strongly nonlinear oscillators are obtained by the Kuzmak-Luke(K-L) method, and the numerical order verification is applied to verify that the asymptotic solutions are valid when the parameter is small but not uniformly valid. The reason is also simply analysized. The error analysis shows that the errors are about one-tenth of the small parameter. Comparisons are also made with different amplitudes to compare the accuracy of the three methods.