# examples/VanDerPolOscillator

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display: latex

$$\begin{array}{rcl} \frac{ d x }{ dt } & = & y \\ \frac{ d y }{ dt } & = & \left(\mu \cdot y \cdot \left(1 - x^{ 2 }\right) - x\right) \\ \end{array}$$
Variable Type Description Value (default) Datatype
$$\mu$$ parameter dampening strength  Real
$$x$$ free  Real
$$y$$ free  Real

The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation-oscillations and are now known as a type of limit cycle in electrical circuits employing vacuum tubes. When these circuits were driven near the limit cycle, they become entrained, i.e. the driving signal pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of Nature that at certain drive frequencies an irregular noise was deterministic chaos.

The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh and Nagumo extended the equation in a planar field as a model for action potentials of neurons. The equation has also been utilised in seismology to model the two plates in a geological fault, and in studies of phonation to model the right and left vocal fold oscillators.

## Modelica Code

model VanDerPolOscillator
parameter Real mu "dampening strength";
Real x;
Real y;
equation
der(x) = y;
der(y) = (((mu * y) * (1 - (x ^ 2))) - x);
end VanDerPolOscillator;


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