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Biryukov equation

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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

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

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]

where exp denotes the exponential function. Here Expression (2) can be used for real and complex values of sk.

The first half-period’s solution at is

Relaxation oscillations F = 4


The second half-period’s solution is

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 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

The integration constants are obtained by the Levenberg–Marquardt algorithm. With , Eq. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.

References

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  1. ^ H. P. Gavin, The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems (MATLAB implementation included)
  2. ^ Arrowsmith D. K., Place C. M. Dynamical Systems. Differential equations, maps and chaotic behavior. Chapman & Hall, (1992)
  3. ^ 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