\chapter{Review of Circuits}
In this chapter we collect facts from circuit theory that we will be using thorought this text. For more information on linear and nonlinear circuits we refer the reader to the book by Chua, Desoer, and Kuh~\cite{chua87}.

\section{The Dynamic Route}
The dynamic route method is a graphical technique that can be used to analyze dynamic behavior of systems modeled by first-order ordinary differential equations of the form
\[
\dot{x}=f(x),\quad x_0=x(t_0).
\]
The method uses a graph of $\dot{x}=f(x)$ versus $x$. Such a graph is referred to as the {\em dynamic route}. An example of a dynamic route is shown in Figure~\ref{fig:dy_rout}.
\begin{figure}
\vspace{3in}
\hskip1.25in\special{bmp:c:/mybook/dy_rout.bmp x=4in y=3in}
\caption{An example of a dynamic route.}
\label{fig:dy_rout}
\end{figure}
Suppose now that we start from the initial condition $x_0=x(0)$. Then, the trajectory $y=x(t)$ that is the solution of the initial value problem
\[
\dot{y}=f(y),\quad y(0)=x_0
\]
will be moving towards $x_1$ because at the point $x_0$ we have
\[
\frac{dx}{dt}<0.
\]
Note that the points $0$, $x_1$, and $x_2$ are equilibrium points because at these points we have
\[
\frac{dx}{dt}=0.
\]
Furthermore the equailibrium points $x_1$ and $x_2$ are asymptotically stable, while $x=0$ is unstable.

To get a better feeling about the use of the dynamic route, we consider a linear system modeled by a first order differential equation
\[
\dot{x}=ax.
\]
In Figure~\ref{fig:dy_lin} we show dynamic routes for the case when $a>0$ and when $a<0$. We can see that when $a>0$ the equilibrium point $x=0$ is unstable, while in the second case when $a<0$ the equilibrium point $x=0$ is asymptotically stable.
\begin{figure}
\vspace{3in}
\hskip1.25in\special{bmp:c:/mybook/dy_lin.bmp x=4in y=3in}
\caption{Dynamic routes of a linear system for $a>0$ and $a<0$.}
\label{fig:dy_lin}
\end{figure}