Biryukov equation explained

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.^[1]

The equation is given by $\frac+f(y)\frac+y=0, \qquad\qquad (1)$

where is a piecewise constant function which is positive, except for small as

$\begin& f(y) = \begin -F, & |y|\le Y_0; \\[4pt] F, & |y|>Y_0. \end \\[6pt]& F = \text > 0, \quad Y_0 = \text > 0. \end$

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]

$y_k(t) = A_\exp(s_t) + A_\exp(s_t) \qquad\qquad (2)$

where denotes the exponential function. Here $s_k = \begin \displaystyle \frac\mp\sqrt, & |y| Expression (2) can be used for real and complex values of .$

The first half-period’s solution at

y(0)=\pmY₀

$\beginy(t) &= \begin y_1(t), & 0\le t$

The second half-period’s solution is

$y(t)= \begin \displaystyle -y_1\left(t-\frac\right), & \displaystyle \frac \le t < \frac + T_0; \\[4pt] \displaystyle -y_2\left(t-\frac\right), & \displaystyle \frac + T_0 \le t < T. \end$

The solution contains four constants of integration, the period and the boundary between and needs to be found. A boundary condition is derived from continuity of and .^[3]

Solution of (1) in the stationary mode thus is obtained by solving a system of algebraic equations as

$\begin& y_1(0) = -Y_0 & y_1(T_0) = Y_0 \\[6pt]& y_2(T_0) = Y_0 & y_2 \! \left(\tfrac\right) = Y_0 \\[6pt]& \displaystyle \left.\frac\right|_ = \left.\frac\right|_ \qquad & \displaystyle \left.\frac\right|_ = -\left.\frac\right|_\frac\end$

The integration constants are obtained by the Levenberg–Marquardt algorithm. With

f(y)=\mu(-1+y²⁾

\mu=const.>0,

Eq. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.

Notes and References

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