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- from D.M. Klimov, V.M. Rudenko, Computer Algebra Methods in
Mechanics, Nauka, (1989).
- nonlinear periodic movement, nonlinear ordinary differential equation
![displaymath11048](img270.gif)
- new time variable
,
is an unknown frequency
![displaymath11049](img273.gif)
- Poincare perturbation method--expansion into a power series
![eqnarray7498](img274.gif)
- example: the van der Pol equation
![eqnarray7502](img275.gif)
- after substituting power series in
![eqnarray7504](img276.gif)
- initial condition
- zeroth order solution
- first order solution
![displaymath11050](img277.gif)
- after Fourier expansion
![eqnarray7506](img278.gif)
- no secular terms in the solution require zero coefficients of
and
on the right hand side
![displaymath11051](img281.gif)
- solution
![eqnarray7514](img282.gif)
- using the initial condition
- second order equation
![eqnarray7518](img283.gif)
- first order solution
![displaymath11052](img284.gif)
- second order solution
![displaymath11053](img285.gif)
- etc. for higher orders
Richard Liska