\chapter{Continuous simulation in detail}
A classic example of a system which can be simulated using differential equations is the size of a given population \citep{birta2013modelling}. Let's say there is an animal population with a size $P(t)$ where $t$ is a certain point in time. This simple definition already highlights the continuous nature of such a model, as the equation simulates the population size for a continuous time domain. With the population size as stated above, the population increase at a given point in time can be formulated using the differential equation $\frac{dP}{dt} = b(t) - d(t)$. In this equation $b(t)$ equates the rate of birth and $d(t)$ the rate of death at time $t$. Using the following system of equations, this model can be elaborated further into a full fledged model:

\begin{align}
    b(t) &= P(t) \cdot 0.5\\
    d(t) &= P(t)^2 \cdot 0.03
\end{align}

Parameters like $0.5$ and $0.03$ as used in the example above, are determined based on experimentation or observations as they occur in the system which is sought to be simulated or based on assumptions about this particular system. Having formulated these equations, their behaviour can be plotted using a vector field. With such a diagram, expected behaviour will become apparent. This is also the case for the example model, as can be seen in its corresponding vector field in figure \ref{plot:ode}, where, for example, a clear equilibrium point can be observed at $P(t) \approx 26.5$. Based on these comparatively simple operations, one is able to predict that, in the case of this example population, any population with an initial size of $P(0) < 26.5$ is expected to increase in size as time passes, whereas a population with an initial size of $P(0) > 26.5$ will shrink according to this model.

\begin{figure}
    \centering
    \includegraphics[width=0.6\textwidth]{figures/ode}
    \caption{Vector plot of the differential equation system as described in section \ref{sec:cts}}
    \label{plot:ode}
\end{figure}

Although the example given above is rather simplistic, continuous models allow for the simulation of systems with more complexity. As long as equations can be devised which accurately describe the desired characteristics of a system, such a system should theoretically be modelable using continuous methods. Herein lies however also one of the disadvantages of continuous models, as the success of a given model greatly depends on the ability to formulate the appropriate equations. This fact is describes in \cite{kohl2016systemc} as it applies to the modelling of hardware using equations.

\subsubsection{Application in hardware modelling}
Knowing this, one might wonder how such techniques could be applied to the simulation of computing hardware, as these components would on a first glace not lend themselves to a behavioural description using mathematical equations. As it tuns out however, behaviour of fundamental electronic components like capacitors, resistors, inductors and voltage sources and the the way these parts interact with each other inside a circuit can also be modelled using mathematical equations. This fact might not be too surprising to anyone with an above-average background in the field of physics as it pertains to electricity. 

An example of this ability to model electrical circuits using continuous modelling is featured in \cite{birta2013modelling}. A circuit, as features in figure \ref{fig:circ-exm}, contains a voltage source ($E(t)$), a resistor ($R$), a capacitor ($C$) and an inductor ($I$) in series. The book formulates an equation (equation \ref{eq:circ}) which is able to model the state of the circuit based on Kirchoff's voltage law. Without getting into extraneous detail about the specifics of electrical engineering, it follows from this equation that the state of the components present inside the circuit can be derived from the initial values of the state variables ($q(t)$ and $q'(t)$ at $t_0$) and the value of the input variable ($E(t)$).

\begin{figure}
    \centering
    \includegraphics[width=0.6\textwidth]{figures/circ-exm.png}
    \caption{Example circuit \citep[p.~251]{birta2013modelling}}
    \label{fig:circ-exm}
\end{figure}

\begin{align}
\label{eq:circ}
    Lq''(t) + Rq'(t) + \frac{q(t)}{C} = E(t)
\end{align}

These general characteristics of electronic components have also been used in a more computational context. In the past, the use of electrical analogue computing equipment, like the machine in figure \ref{fig:eai680} were used to perform simulations on physics models---e.g. tidal calculations or certain models used by the oil industry. These analogue computers contain circuits with basic electrical components whose electrical properties are similar to certain mathematical constructs. As such, a circuit built from the appropriate components is able to be used as the electrical analogue of an equation. When looking at the simulation of such machines, we are presented with a peculiar chicken and egg problem. It is clear that the analogue machine in this example was used for continuous simulation purposes, and as such, simulating it using a continuous model should logically be possible. While no example of such a simulator appears to exist, is need be, it should be possible to device a model to simulate the functionality of such a computer.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.6\textwidth]{figures/eai680.jpg}
    \caption{Electronic Associates Inc. Model 680 \citep{uva_computermuseum}}
    \label{fig:eai680}
\end{figure}