``MAPLE'' analysis of nonlinear oscillations (Q1206169)

Two MAPLE programs (MAPLE is a symbolic manipulation computer language) for the application of the intrinsic harmonic balancing (IHB) technique (a perturbation method) to the analysis of nonlinear autonomous and nonautonomous oscillations are presented. The first program computes approximations for periodic solutions of the form \[ x(t,\varepsilon)=p_ 0(\varepsilon)+\sum_{m=1}^ M (p_ m(\varepsilon)\cos m\omega(\varepsilon)t+r_ m(\varepsilon)\sin m\omega(\varepsilon)t) \] for autonomous systems \(\ddot x+kx+\varepsilon f(x,\dot x)=0\), \(\varepsilon\) small. The second program finds approximate periodic solutions \[ x(t,\varepsilon)=p_ 0(\varepsilon)+\sum_{m=1}^ M (p_ m(\varepsilon)\cos m\Omega t+r_ m(\varepsilon)\sin m\Omega t) \] of periodically excited systems \(\ddot x+d\dot x+kx+\varepsilon f(x,\dot x)=C_ 1 \cos \Omega t+ C_ 2\sin \Omega t\) with small \(\varepsilon\). The approximations can be performed up to an order \(n\) in \(\varepsilon\). Three illustrative examples (Duffing-, van der Pol- and forced Duffing- oscillator) are given.