where ƒ(y) is a piecewise constant function which is positive, except for small y as
Eq. (1) is a special case of the Lienard equation; it describes the auto-oscillations.
Solution (1) at a separate time intervals when f(y) is constant is given by[2]
where exp denotes the exponential function. Here
Expression (2) can be used for real and complex values of sk.
The first half-period’s solution at is
The second half-period’s solution is
The solution contains four constants of integrationA1, A2, A3, A4, the period T and the boundary T0 between y1(t) and y2(t) needs to be found. A boundary condition is derived from continuity of y(t) and dy/dt.[3]
Solution of (1) in the stationary mode thus is obtained by solving a system of algebraic equations as
^H. P. Gavin, The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems (MATLAB implementation included)
^Arrowsmith D. K., Place C. M. Dynamical Systems. Differential equations, maps and chaotic behavior. Chapman & Hall, (1992)
^Pilipenko A. M., and Biryukov V. N. «Investigation of Modern Numerical Analysis Methods of Self-Oscillatory Circuits Efficiency», Journal of Radio Electronics, No 9, (2013). http://jre.cplire.ru/jre/aug13/9/text-engl.html