## 1pPA11. An analytical radial solution to O((cursive beta)[sup 4]) of the Van der Pol--Rayleigh limit cycle oscillator.

### Session: Monday Afternoon, December 2

### Time: 4:30

**Author: David T. Raphael**

**Location: Dept. of Anesthesiol., Texas Tech. Univ., Health Sci. Ctr., 4800 Alberta Ave., El Paso, TX 79905**

**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.

ASA 132nd meeting - Hawaii, December 1996