Biryukov equation
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.
{\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(y){\frac {dy}{dt}}+y=0,\qquad \qquad (1)}
where ƒ(y) is a piecewise constant function which is positive, except for small y as
{\displaystyle {\begin{aligned}&f(y)={\begin{cases}-F,&|y|\leq Y_{0};\\[4pt]F,&|y|>Y_{0}.\end{cases}}\\[6pt]&F={\text{const.
(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[2]
exp (
exp (
where exp denotes the exponential function.
{\displaystyle s_{k}={\begin{cases}\displaystyle {\frac {F}{2}}\mp {\sqrt {\left({\frac {F}{2}}\right)^{2}-1}},&|y| }}\end{cases}}} Expression (2) can be used for real and complex values of sk. The first half-period’s solution at ⋅ exp ( ⋅ exp ( ⋅ exp ( ⋅ exp ( {\displaystyle {\begin{aligned}y(t)&={\begin{cases}y_{1}(t),&0\leq t The second half-period’s solution is {\displaystyle y(t)={\begin{cases}\displaystyle -y_{1}\left(t-{\frac {T}{2}}\right),&\displaystyle {\frac {T}{2}}\leq t<{\frac {T}{2}}+T_{0};\\[4pt]\displaystyle -y_{2}\left(t-{\frac {T}{2}}\right),&\displaystyle {\frac {T}{2}}+T_{0}\leq t The solution contains four constants of integration A1, 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 the 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 {\displaystyle {\begin{array}{ll}&y_{1}(0)=-Y_{0}&y_{1}(T_{0})=Y_{0}\\[6pt]&y_{2}(T_{0})=Y_{0}&y_{2}\!\left({\tfrac {T}{2}}\right)=Y_{0}\\[6pt]&\displaystyle \left. {\frac {dy_{1}}{dt}}\right|_{T_{0}}=\left. {\frac {dy_{2}}{dt}}\right|_{T_{0}}\qquad &\displaystyle \left. {\frac {dy_{2}}{dt}}\right|_{\frac {T}{2}}\end{array}}} The integration constants are obtained by the Levenberg–Marquardt algorithm. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.
