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Non-linear second-order differential equation
Sine oscillations F = 0.01
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
d
2
y
d
t
2
+
f
(
y
)
d
y
d
t
+
y
=
0
,
(
1
)
{\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
f
(
y
)
=
{
−
F
,
|
y
|
≤
Y
0
;
F
,
|
y
|
>
Y
0
.
F
=
const.
>
0
,
Y
0
=
const.
>
0.
{\displaystyle {\begin{aligned}&f(y)={\begin{cases}-F,&|y|\leq Y_{0};\\[4pt]F,&|y|>Y_{0}.\end{cases}}\\[6pt]&F={\text{const.}}>0,\quad Y_{0}={\text{const.}}>0.\end{aligned}}}
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
1
,
k
exp
(
s
1
,
k
t
)
+
A
2
,
k
exp
(
s
2
,
k
t
)
(
2
)
{\displaystyle y_{k}(t)=A_{1,k}\exp(s_{1,k}t)+A_{2,k}\exp(s_{2,k}t)\qquad \qquad (2)}
where exp denotes the exponential function . Here
s
k
=
{
F
2
∓
(
F
2
)
2
−
1
,
|
y
|
<
Y
0
;
−
F
2
∓
(
F
2
)
2
−
1
otherwise.
{\displaystyle s_{k}={\begin{cases}\displaystyle {\frac {F}{2}}\mp {\sqrt {\left({\frac {F}{2}}\right)^{2}-1}},&|y|<Y_{0};\\[2pt]\displaystyle -{\frac {F}{2}}\mp {\sqrt {\left({\frac {F}{2}}\right)^{2}-1}}&{\text{otherwise.}}\end{cases}}}
Expression (2) can be used for real and complex values of
sk .
The first half-period’s solution at
y
(
0
)
=
±
Y
0
{\displaystyle y(0)=\pm Y_{0}}
is
Relaxation oscillations F = 4
y
(
t
)
=
{
y
1
(
t
)
,
0
≤
t
<
T
0
;
y
2
(
t
)
,
T
0
≤
t
<
T
2
.
y
1
(
t
)
=
A
1
,
k
⋅
exp
(
s
1
,
k
t
)
+
A
2
,
k
⋅
exp
(
s
2
,
k
t
)
,
y
2
(
t
)
=
A
3
,
k
⋅
exp
(
s
3
,
k
t
)
+
A
4
,
k
⋅
exp
(
s
4
,
k
t
)
.
{\displaystyle {\begin{aligned}y(t)&={\begin{cases}y_{1}(t),&0\leq t<T_{0};\\[4pt]y_{2}(t),&\displaystyle T_{0}\leq t<{\frac {T}{2}}.\end{cases}}\\[4pt]y_{1}(t)&=A_{1,k}\cdot \exp(s_{1,k}t)+A_{2,k}\cdot \exp(s_{2,k}t),\\[2pt]y_{2}(t)&=A_{3,k}\cdot \exp(s_{3,k}t)+A_{4,k}\cdot \exp(s_{4,k}t).\end{aligned}}}
The second half-period’s solution is
y
(
t
)
=
{
−
y
1
(
t
−
T
2
)
,
T
2
≤
t
<
T
2
+
T
0
;
−
y
2
(
t
−
T
2
)
,
T
2
+
T
0
≤
t
<
T
.
{\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<T.\end{cases}}}
The solution contains four constants of integration A 1 , A 2 , A 3 , A 4 , the period T and the boundary T 0 between y 1 (t ) and y 2 (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
y
1
(
0
)
=
−
Y
0
y
1
(
T
0
)
=
Y
0
y
2
(
T
0
)
=
Y
0
y
2
(
T
2
)
=
Y
0
d
y
1
d
t
|
T
0
=
d
y
2
d
t
|
T
0
d
y
1
d
t
|
0
=
−
d
y
2
d
t
|
T
2
{\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_{1}}{dt}}\right|_{0}=-\left.{\frac {dy_{2}}{dt}}\right|_{\frac {T}{2}}\end{array}}}
The integration constants are obtained by the Levenberg–Marquardt algorithm .
With
f
(
y
)
=
μ
(
−
1
+
y
2
)
{\displaystyle f(y)=\mu (-1+y^{2})}
,
μ
=
const.
>
0
,
{\displaystyle \mu ={\text{const.}}>0,}
Eq. (1) named Van der Pol oscillator . Its solution cannot be expressed by elementary functions in closed form.
References [ edit ]
^ 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