Tuesday, February 2, 2010

Van Der Pol System

vanderPolEquation

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting y=y^'. It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by

 y^('')-mu(1-y^2)y^'+y=0.

If mu=0, the equation reduces to the equation of simple harmonic motion

 y^('')+y=0.

The van der Pol equation is

[Graphics:Images/VanDerPolMod_gr_1.gif],
where
[Graphics:Images/VanDerPolMod_gr_2.gif] is a constant.

When [Graphics:Images/VanDerPolMod_gr_3.gif] the equation reduces to [Graphics:Images/VanDerPolMod_gr_4.gif], and has the familiar solution [Graphics:Images/VanDerPolMod_gr_5.gif]. Usually the term [Graphics:Images/VanDerPolMod_gr_6.gif] in equation (1) should be regarded as friction or resistance, and this is the case when the coefficient [Graphics:Images/VanDerPolMod_gr_7.gif] is positive. However, if the coefficient [Graphics:Images/VanDerPolMod_gr_8.gif] is negative then we have the case of "negative resistance." In the age of "vacuum tube" radios, the "tetrode vacuum tube" (cathode, grid, plate), was used for a power amplifier and was known to exhibit "negative resistance." The mathematics is amazing too, and van der Pol, Balthasar (1889-1959) is credited with developing equation (1). The solution curves exhibits orbital stability. The van der Pol equation can be written as a second order system

[Graphics:Images/VanDerPolMod_gr_9.gif],
and
[Graphics:Images/VanDerPolMod_gr_10.gif].

Any convenient numerical differential equation solver such as the
Runge-Kutta method can be used to compute the solutions.

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Introductory Methods of Numerical Analysis