@ -122,6 +122,11 @@ If the system is perturbed slightly, it always returns to the standard cycle. Li

\label{f:lc_hst}

\end{figure}

\subsection{Damping}

Mentioning damping is important mainly because, in a real world, oscillations eventually stop, due to Newton's law of Thermodynamics (the frictional force). In electronics, there is no ideal oscillator, too - small amount of energy is lost every cycle, due to electric resistance.

Generally, the damping is linear either linear or non-linear. As a rule of thumb, the linear one is easily modeled mathematically, obeying known rules, while the non-linear one is not \cite{institute1989estimation}. There are some use cases, where non-linear damping is advantageous, but the research is still ongoing about this topic.

\subsection{Li\'{e}nard Equation}

A non-linear second-order ordinary differential equation

where $y$ is the dynamical variable and $\mu>0$ is a parameter. If $\mu=0$, then the equation reduces to the equation of simple harmonic motion

$$y''+y=0$$

The $\mu$ parameter determines the shape of the limit cycle. As it approaches 0, it gets the shape of a circle. On the other hand, increasing the paramter, involves sharpening of the curves.

The Van der Pol equation \eqref{eq:vdp} arises in the study of circuits containing vacuum tubes (triode) and is derived from earlier, Rayleigh equation \cite{nahin2001science}, known also as Rayleigh-Plesset equation - an ordinary differential equation explaining the dynamics of a spherical bubble in an infinite body of liquid.

@ -165,10 +171,12 @@ However, this form is not well-known. Far common form uses the transformation $y

\end{align*}

which can be plotted onto direction field, as shown on \cref{f:vdp_m}. It is possible to see the stable limit cycle as well as trajectories from both sides attracted towards it.

The Van der Pol oscillator can be forced too, however, this work does not aim to investigate further in this direction.

\begin{figure}[t!]

\centering

\includegraphics[width=.85\linewidth]{vdp_maxima}

\caption{Van der Pol oscillator plotted on the direction field. Parameter $\mu=1$. The wxMaxima computing software was used for this purpose. }

\caption{Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle and the direction field Parameter $\mu=1$. The wxMaxima computing software was used for this purpose. }