In the study of dynamical systems, the Biryukov equation (or Biryukov oscillator), named after Vadim Biryukov (1946), is a non-linear second-order differential equation used to model damped oscillators.
The equation is given by
where is a piecewise constant function which is positive, except for small as
Eq. (1) is a special case of the Lienard equation; it describes the auto-oscillations.
Solution (1) at separate time intervals when f(y) is constant is given by
where denotes the exponential function. Here
Expression (2) can be used for real and complex values of .
The first half-periodâÂÂs solution at is
The second half-periodâÂÂs solution is
The solution contains four constants of integration , the period and the boundary between and needs to be found. A boundary condition is derived from the continuity of and .
Solution of (1) in the stationary mode thus is obtained by solving a system of algebraic equations as
The integration constants are obtained by the LevenbergâÂÂMarquardt algorithm. With , Eq. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.