Biryukov equation

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]

Sine oscillations F = 0.01

The equation is given by

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(y){\frac {dy}{dt}}+y=0,\qquad \qquad (1)}$

where ƒ(y) is a piecewise constant function which is positive, except for small y as

${\displaystyle f(y)={\begin{cases}-F,&|y|\leq Y_{0};\\F,&|y|>Y_{0}.\end{cases}}}$

${\displaystyle F={\text{constant}}>0,\quad Y_{0}={\text{constant}}>0.}$

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]

${\displaystyle y_{k}(t)=A_{1,k}\exp(s_{1,k}t)+A_{2,k}\exp(s_{2,k}t)\qquad \qquad (2)}$

Here ${\displaystyle s_{k}=F/2\mp {\sqrt {(F/2)^{2}-1}}}$ , at ${\displaystyle |y|<Y_{0}}$ and ${\displaystyle s_{k}=-F/2\mp {\sqrt {(F/2)^{2}-1}}}$ otherwise. Expression (2) can be used for real and complex values of ${\displaystyle s_{k}}$.

The first half-period’s solution at ${\displaystyle y(0)=\pm Y_{0}}$ is

Relaxation oscillations F = 4

${\displaystyle y(t)={\begin{cases}y_{1}(t),&0\leq t<T_{0};\\y_{2}(t),&T_{0}\leq t<T/2.\end{cases}}}$

${\displaystyle y_{1}(t)=A_{1,k}\cdot \exp(s_{1,k}t)+A_{2,k}\cdot \exp(s_{2,k}t),}$

${\displaystyle y_{2}(t)=A_{3,k}\cdot \exp(s_{3,k}t)+A_{4,k}\cdot \exp(s_{4,k}t).}$

The second half-period’s solution is

${\displaystyle y(t)={\begin{cases}-y_{1}(t-T/2),&T/2\leq t<T/2+T_{0};\\-y_{2}(t-T/2),&T/2+T_{0}\leq t<T.\end{cases}}}$

The solution contains four constants of integration ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, ${\displaystyle A_{3}}$, ${\displaystyle A_{4}}$, the period ${\displaystyle T}$ and the boundary ${\displaystyle T_{0}}$ between ${\displaystyle y_{1}(t)}$ and ${\displaystyle y_{2}(t)}$ needs to be found. A boundary condition is derived from continuity of ${\displaystyle y(t)}$) and ${\displaystyle dy/dt}$.[3]

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

${\displaystyle y_{1}(0)=-Y_{0}}$; ${\displaystyle y_{1}(T_{0})=Y_{0}}$; ${\displaystyle y_{2}(T_{0})=Y_{0}}$; ${\displaystyle y_{2}(T/2)=Y_{0}}$; ${\displaystyle \left.{\begin{matrix}dy_{1}/dt\end{matrix}}\right|_{T_{0}}=\left.{\begin{matrix}dy_{2}/dt\end{matrix}}\right|_{T_{0}}}$;${\displaystyle \left.{\begin{matrix}dy_{1}/dt\end{matrix}}\right|_{0}=-\left.{\begin{matrix}dy_{2}/dt\end{matrix}}\right|_{T/2}\;\;\;(3)}$.

The integration constants are obtained by the Levenberg–Marquardt algorithm. With ${\displaystyle f(y)=\mu (-1+y^{2})}$, ${\displaystyle \mu =const>0}$, Eq. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.