\IEEEPARstart{T}{his} this paper is intended to sum up the research done in order to understand the Dynamics in electrical systems and their underlying differential equations.

For the most part, applications fall into three broad categories: predictive (also referred to as generative), in which the objective is to predict future states of the system from observations of the past and present states of the system, diagnostic, in which the objective is to infer what possible past states of the system might have led to the present state of the system (or observations leading up to the present state), and, finally, applications in which the objective is neither to predict the future nor explain the past but rather to provide a theory for the physical phenomena. These three categories correspond roughly to the need to predict, explain, and understand physical phenomena.

\subsection{Differential Equations}

A \textbf{differential equation} is any equation which contains derivatives, either ordinary derivatives or partial derivatives. Almost every physical situation that occurs in nature can be \textit{described} with an appropriate differential equation.

A \textbf{differential equation} is any equation which contains derivatives, either ordinary derivatives or partial derivatives. Almost every physical situation that occurs in nature can be \textit{described} with an appropriate differential equation.

The process of describing a physical situation with a differential equation is called \textbf{modeling}.

Differential equations are generally concerned about three questions:

\begin{enumerate}

\item Given a differential equation will a solution exist?

\item Given a differential equation will a solution exist?

\item If a differential equation does have a solution how many solutions are there?

\item If a differential equation does have a solution can we find it?

\end{enumerate}

@ -108,7 +107,7 @@ Differential equations fall to two groups - \textit{ordinary differential equati

\subsection{Direction Field}

Understanding \textbf{direction fields} (or \textbf{slope fields)} and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done before the getting to actually solving them.

Understanding \textbf{direction fields} (or \textbf{slope fields)} and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done before the getting to actually solving them.

The direction fields are important because they can provide a \textit{sketch of solution}, if exist, and a \textit{long term behavior} - most of the time we are interested in general picture about what is happening, as the time passes.

@ -118,7 +117,7 @@ Example direction field, embedded in phase portrait is shown in \cref{f:vdp_m}.

\subsection{Laplace Transform}

The \textbf{Laplace transform} is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform \eqref{eq:lpl} is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The Laplace transform $\mathcal{L}$

where $f(t)$ is defined for $t\le0$ - this is it's most common form and is called \textit{unilateral}.

@ -127,7 +126,7 @@ Most important properties of Laplace transform is that differentiation and integ

\section{Periodic Orbits}

A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an \textit{oscillator}.

A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an \textit{oscillator}.

\subsection{Limit Cycle}

A \textbf{limit cycle} is an isolated closed trajectory. \textit{Isolated} means that neighboring trajectories are not closed - they spiral either towards or away from the limit cycle. The particle on the limit cycle, appears after one period on the exact same spot. Limit cycle appears on on a plane, opposed to a periodic orbit, that happens to be a vector.

@ -139,7 +138,7 @@ A \textbf{limit cycle} is an isolated closed trajectory. \textit{Isolated} means

\label{f:lc_st}

\end{figure}

If all neighboring trajectories approach the limit cycle, we say the limit cycle is \textbf{stable} or \textit{attracting}, as shown on \cref{f:lc_st}. Otherwise the limit cycle is \textbf{unstable}, or in exceptional cases, \textbf{half-stable}. Stable limit cycles are very important scientifically as they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing.

If all neighboring trajectories approach the limit cycle, we say the limit cycle is \textbf{stable} or \textit{attracting}, as shown on \cref{f:lc_st}. Otherwise the limit cycle is \textbf{unstable}, or in exceptional cases, \textbf{half-stable}. Stable limit cycles are very important scientifically as they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing.

\begin{figure}[ht!]

\centering

@ -276,7 +275,7 @@ The RL circuit shown on \cref{f:rl} has a resistor and an inductor connected in

Applying the KVL, we obtain the algebraic sums of all the voltage drops as an ODE with respect to time

$$Ri+L\,\frac{di}{dt}=V(t)$$

solving which we obtain

solving which we obtain

$$i=\frac{V}{R}\left(1-e^{-(R/L)t}\right)$$

The solving process is quite lenghty and is not a point of this work. For more details see \cite{bird2014electrical}.

@ -352,7 +351,7 @@ The main goal was to get some general idea about the connections between the ter

%\appendices

%\section{Proof of the First Zonklar Equation}

%Appendix one text goes here.

%Appendix one text goes here.

%

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@ -397,7 +396,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w

%\begin{figure}[!t]

%\centering

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%\caption{Simulation Results.}

@ -413,7 +412,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w

% The subfigure \label commands are set within each subfloat command,

% and the \label for the overall figure must come after \caption.

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@ -432,7 +431,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w

% reference/describe all of them (a), (b), etc., within the main caption.

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@ -657,8 +656,8 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w