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Limit CirclesBrown University, Applied Mathematics |
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Limit Circles
Example 2.2.3:
The van der Pol oscillator was originally proposed in 1927 by the Dutch electrical engineer and physicist Balthasar van der Pol (1889--1959) while he was working at Philips. It is used to model a non-conservative oscillator with non-linear damping according to the second-order differential equation:
\[
\frac{{\text d}^2 x}{{\text d} t^2} - \epsilon \left( 1- x^2 \right) \frac{{\text d} x}{{\text d} t} + x =0 ,
\]
where x is the position coordinate—which is a function of the time t, and \( \epsilon \) is a scalar parameter indicating the nonlinearity and the strength of the damping. Since energy is dissipated at high amplitudes and generated at low amplitudes, there exists oscillations around a state at which energy generation and dissipation balance. The state towards which the oscillations converge is known as a limit cycle. A limit cycle is an isolated closed trajectory.
Balthazar van der Pol was a pioneer in the field of radio and telecommunications. 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. Van der Pol equation can be transferred to a system of first order differential equations:
\[
\begin{split}
\dot{x} &=y, \\
\dot{y} &= x -\epsilon \left( 1- x^2 \right) y. \end{split}
\]
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