Abstract:
An analytical closed-form solution to O((cursive beta)[sup 4]) of the Van der Pol--Rayleigh limit cycle oscillator is derived assuming a constant angular frequency as given by the Davis--Alfriend solution, with (cursive beta)(less than or equal to)1. Using phase plane polar coordinates, the Van der Pol equation is transformed into a time-dependent radial equation. A variation-of-parameters approach is then employed to derive an exact infinite series expression involving modified Bessel functions and sinusoidal functions. The series is then simplified using a combination of a Sommerfeld--Watson technique in conjunction with a series reversal via an Euler transformation. The solution for the limit cycle radius can be reduced to a relatively simple mathematical form. The computed limit cycle paths with this radial solution correspond very closely to the true solution paths for (cursive beta)(less than or equal to)1 and are demonstrated for comparison. Extension of this technique to the problem of large (cursive beta) is discussed.