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}.

@ -123,7 +123,7 @@ 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 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

@ -208,6 +208,8 @@ The Van der Pol oscillator can be forced too, however, this work does not aim to

\section{Applications in Electronics}

%The modeling met some more advanced applications in electronics field, too. We will briefly summarize some of them, in this section.

\subsection{Josephson Junctions}

\textbf{Josephson junctions} are superconducting devices that are capable of generating voltage oscillations of extraordinary high frequency, typically 10\textsuperscript{10} - 10\textsuperscript{11} cycles per second \cite{van1981principles}. They consist of two superconducting layers, separated by a very thin insulator that weakly couples them, as shown on \cref{f:jjunc}.

@ -253,16 +255,13 @@ Appendix two text goes here.

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\section*{Acknowledgment}

The authors would like to thank...

The authors would like to thank professor Carlos Par\'{e}s for having patience with them. Another thank would go to the well-written book \emph{Nonlinear Dynamics and Chaos, S.H. Strogatz, 2008} for introduction to and for sparking curiosity in the field of Dynamical Systems.

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