Van Der Pol was solved using perturbation for first order approximation. It was solved using two methods. In the first method, the solution was restricted to not include forcing functions which leads to resonance.
In the second method, no such restriction was made. In both methods, this problem was solved for general initial conditions (i.e. both \(x\left ( 0\right ) \) and \(\dot {x}\left ( 0\right ) \) can both be nonzero.
In the first method, it was found that the we must restrict the initial conditions to be such that \(x\left ( 0\right ) ^{2}\) \(+\dot {x}\left ( 0\right ) ^{2}=4\). This is the condition which resulted in the resonance terms vanishing from the solution. This leads to a solution which generated a limit cycle which did not blow up as the solution is run for longer time. In other words, once the solution enters a limit cycle, it remains in the limit cycle.
In the second method, no restriction on the initial conditions was made. This allowed the solution to start from any state. However, the limit cycle would grow with time, and the solution will suffer from fluttering due to the presence of the resonance terms. However, even though the solution was not stable in the long term, this second approach allowed one to examine the solution for a shorter time but with the flexibility of choosing any initial conditions.
It was also observed that increasing the value of the perturbation parameter \(\alpha \) gradually resulted in an inaccurate solution as would be expected, as the solutions derived here all assumed a very small1 \(\alpha \) was used in generating the solutions and plots.
For future research, it would be interesting to consider the effect of higher order approximation on the solutions and compare with accurate numerical solutions.