From 62605391e53058ea3e381abbb116fcde1b5dd4e7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Peter=20Babi=C4=8D?= Date: Sun, 7 Jun 2015 18:30:20 +0200 Subject: [PATCH] two-dimensional transformation --- differential-equations/electrical.bib | 23 ++++++++- differential-equations/electrical.pdf | Bin 102582 -> 141282 bytes differential-equations/electrical.tex | 47 ++++++++++++++---- differential-equations/figures/vdp_maxima.png | Bin 0 -> 18998 bytes 4 files changed, 59 insertions(+), 11 deletions(-) create mode 100644 differential-equations/figures/vdp_maxima.png diff --git a/differential-equations/electrical.bib b/differential-equations/electrical.bib index a49293a..383d3e8 100644 --- a/differential-equations/electrical.bib +++ b/differential-equations/electrical.bib @@ -21,7 +21,7 @@ @book{oscillations, title={Oscillations and Waves}, - author={K, G.C. and Sanjay, gupta and Suresh, garg}, + author={Garg, S. and Gosh, C.K. and Gupta, S.}, isbn={9788120339217}, url={https://books.google.es/books?id=45SUdYy1v8sC}, publisher={Prentice-Hall Of India Pvt. Limited} @@ -38,3 +38,24 @@ publisher={Springer New York}, pages=96 } + +@book{sternberg2014dynamical, + title={Dynamical Systems}, + author={Sternberg, S.}, + isbn={9780486135144}, + url={https://books.google.es/books?id=9SL0AwAAQBAJ}, + year={2014}, + publisher={Dover Publications}, + pages=211 +} + +@book{kaplan2012understanding, + title={Understanding Nonlinear Dynamics}, + author={Kaplan, D. and Glass, L.}, + isbn={9781461208235}, + series={Textbooks in Mathematical Sciences}, + url={https://books.google.es/books?id=gh\_vBwAAQBAJ}, + year={2012}, + publisher={Springer New York}, + pages={240--244} +} diff --git a/differential-equations/electrical.pdf b/differential-equations/electrical.pdf index bbcf6a702cb83e86366a720724809a2f35163190..295234499d50a210ad7725aee4caf05b306d6a18 100644 GIT binary patch delta 116295 zcmZ6y2UJsC6D~{(NR1#xx}qQ*5s)SzC?Z8sG$9lr(v;po>Iq${N>vb$E=hpUJ0v2~ 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Under certain additional assumptions Li\'{e}nard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. \subsection{Van der Pol Oscillator} -The limit cycle property translated to real life application means oscillations. Oscillations are important part of electronics \cite{oscillations}. One of the most well-known oscillator model in dynamics is \textbf{Van der Pol oscillator}, described by differential equation +One of the most well-known oscillator model in dynamics is \textbf{Van der Pol oscillator}, which is a special case of Li\'{e}nard's equation \eqref{eq:lnrd} and is described by a differential equation \begin{equation} \label{eq:vdp} y''-\mu\left(1-y^2\right)y'+y=0 @@ -137,13 +147,30 @@ y''-\mu\left(1-y^2\right)y'+y=0 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 Van der Pol equation \eqref{eq:vdp} arises in the study of circuits containing vacuum tubes 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. +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. Van der Pol oscillator is \textbf{self-sustainable}, \textbf{relaxation} oscillator. Self-sustainability in this context means, that the energy is fed into small oscillations and removed from large oscillations. Relaxation means, that the energy is gradually accumulating over time and then quickly released (relaxed). In electronics jargon, the relaxation oscillator is also called a \textit{free-running} oscillator. As already explained, it does not require neither one (monostable), nor two (bistable) inputs for transitioning between states, it "runs" by itself, thus free-running. - - - +\subsection{Van der Pol's Equation Limit Cycle} +Li\'{e}nard's theorem can be used to prove that the system described by Van der Pol equation \eqref{eq:vdp} has a limit cycle \cite{sternberg2014dynamical}. If we want to visualize it, the one-dimensional form of equation must be first \textit{transformed} to the two-dimensional form. Applying the Li\'{e}nard transformation $$y=x-\frac{x^3}{3}-\frac{\dot x}{\mu}$$ where dot indicates the time derivative, the system can be written in it's two-dimensional form \cite{kaplan2012understanding}: +\begin{align*} +\dot x &= \mu \left(x-\frac13 x^3 -y\right) \\ +\dot y &= \frac{1}{\mu} x +\end{align*} + +However, this form is not well-known. Far common form uses the transformation $y=\dot x$, that yields +\begin{align*} +\dot x &= y \\ +\dot y &= \mu\left(1-x^2\right)y-x +\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. + +\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. } + \label{f:vdp_m} +\end{figure} \section{Conclusion} diff --git a/differential-equations/figures/vdp_maxima.png b/differential-equations/figures/vdp_maxima.png new file mode 100644 index 0000000000000000000000000000000000000000..a359f81cec6999351bc69e3967ff0b35029b4f3b GIT binary patch literal 18998 zcma&ObyQVv*EYJgbZtP8ZV&{vv~)L80)m@THjRWxr*x>4q;z*}QURqKDM31=VIvLF zXYqTU=l$OIJ!g#b@eg~fvDUrrnD@M|dChCC_s=wx2ym%z0RSLSQI^*RfO}#9fJTLl z30^VL3iAQ~p*zc}JjVwAe6Y>K0e}Hek(YVyk+wT;-^)5_+;ebiCwSQ&Za%1Iie2|Q z=RSL0HN`;3K|q)9Cn{3y$x>?)#-i2bkTz-bY({LxuVc?-bh^wVAF{9q1lNaB z81ZbqR=%{vQ(=Mk>5%K&9Pu8PdUJak2eROl)^Gz3Vrh7M;%+If& ztD6A!Ot?;|3H%2z61xF_zilur04z}w z;^!SwS90e!XD!@>62~JmewW)XS~wQ3|G2N6z1us(76ot3E9VqrVtUhZU2o*Im4Emt z+DE?mD1EQjb0eLNiK*qpu;t;a^ZpmBVvFZdSDl(y^84-g03hPYrb8M1^(cM9;?3@2 z;^a$j^$d^SIN(5b%D=3>XUmAu^t%{C2Pea1V##p$b~E}$Szf-snw0p{-58SSua>av z)W8KUAZU!xybcte{Jd!Ur$Ac!05qo z!g3QWPXEk%yG0LW%Z2?j^lzh+St%Ps)Q!3I|u+XV>;s 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