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Solving the Van Der Pol nonlinear differential equation using first order approximation perturbation method

Nasser M. Abbasi

Sometime in 2009   Compiled on October 27, 2025 at 8:19pm

Abstract

The Van Der Pol differential equation

\[ x''(t) -\alpha \left ( 1-x^2(t) \right ) x'(t) +x(t) = 0 \]
Was solved using perturbation with first order approximation. Two different solutions were obtained. The first solution restricted the initial conditions to be \(x(0)^{2}+x^{\prime 2}=4\) which resulted in forcing function that caused resonance to be eliminated. This gave a stable solution but with initial conditions restricted to be near the origin of the phase plane space.

The second solution allowed arbitrary initial conditions any where in the phase plane but the the resulting forcing function caused resonance resulting in a solution which became unstable after some time.

Phase plane plots are used to compare the two solutions.

1 First solution: Restriction on initial condition. No resonance
2 Second solution. No restriction on initial conditions. Resonance present in the solution
3 Conclusion