Mechanics - nonlinear ordinary differential equations

 

• from D.M. Klimov, V.M. Rudenko, Computer Algebra Methods in Mechanics, Nauka, (1989).

• nonlinear periodic movement, nonlinear ordinary differential equation

  
  
  

• new time variable , is an unknown frequency

  
  
  

• Poincare perturbation method--expansion into a power series

  
  
  

• example: the van der Pol equation

  
  
  

• after substituting power series in

  
  
  

• initial condition

• zeroth order solution

• first order solution

  
  
  

• after Fourier expansion

  
  
  

• no secular terms in the solution require zero coefficients of and on the right hand side

  
  
  

• solution

  
  
  

• using the initial condition

• second order equation

  
  
  

• first order solution

  
  
  

• second order solution

  
  
  

• etc. for higher orders