Dr. Robert Lopez

A Striking Difference Between Linear and Nonlinear Oscillators

There is a remarkable difference between the frequencies of the two oscillators ẍ + k2 x = 0 and ẍ + x + sin(a x) = 0, (the first linear; the second, nonlinear), both subject to the initial conditions x(0) = A, ẋ(0) = 0. The frequency of the linear oscillator does not depend on the initial amplitude A, but that of the nonlinear oscillator (where a is a system parameter), does. It is easy to see the frequency independence in the linear case via the explicit closed-form solution that is readily available. Unfortunately, there is no such solution in the nonlinear case.

To study the dependence of the frequency on the initial amplitude for the nonlinear oscillator, various analytic approximations schemes must be invoked - typically, a Lindstedt perturbation.

Recently, Maple added a command for generating such a perturbation expansion, and included it in the Student ODEs package where it inherits the ability to provide the steps of the calculation. We'll demonstrate this functionality, and for added insight also obtain the expansion without this package, ending up with analytic and visual evidence of the dependence of the frequency on the initial amplitude for the nonlinear oscillator.
Language: English
Duration: 51 Minutes
Related Terms: Oscillator, Ode, Linear, Nonlinear