We summarize the salient points of this chapter.

We first considered some examples of one-dimensional differential equations, that is equations of the form $\dot{x}=f(x)$, where $x\in\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$. We expect that a wealth of informations about the behavior of solutions to be easily obtainable from a simple graphical study, namely by plotting $\dot{x}$ versus $x$, without actually solving the equation. What was left is the question of existence and uniqueness of solutions and a systematic treatment of fixed points.

We then considered differential equations in the plane, that is, equations of the form

$$ \begin{cases} \dot{x}=f(x,y)\\ \dot{y}=g(x,y) \end{cases} $$

where $(x,y)\in\mathbb{R}^2$ and $f,g:\mathbb{R}^2\to\mathbb{R}^2$. The first example was Volterra’s model of interaction between sharks and sardines. We understood how to visualize at once all trajectories in the $xy$-plane to get the `phase portrait’ of the equations. A trajectory can be thought of as the curve that a particle would trace out when dropped into the flow of an imaginary fluid whose velocity vector is given by $(f(x,y),g(x,y))$ at each point $(x,y)$. Again, we left aside the question of existence and uniqueness of solutions.Population models like Volterra’s one are examples of first-order differential equations since the unknown functions $x(t)$ and $y(t)$ are determined by equations involving their first derivative.

Examples coming from classical mechanics are second-order differential equations. Indeed, if one considers for instance a particle of mass $m$ constrained to a straight line, its motion is governed, according to Newton’s second law, by an equation of the form $m\ddot{x}=f(x)$ where $f(x)$ is the force acting at point $x$. Thus, the unknown function $x(t)$ is determined by an equation involving its second derivative. We saw a very nice trick to visualize in the $xy$-plane this kind of equation by setting $y=\dot{x}$, which gives $\dot{y}=\ddot{x}=f(x)$; this leads to the equations

$$ \begin{cases} \dot{x}=y\\ \dot{y}=\frac{f(x)}{m} \cdot \end{cases} $$

We thus treat velocity as a second dimension. We can then interpret the equation $m\ddot{x}=f(x)$ as an imaginary fluid flow in the $xy$-plane whose velocity field at $(x,y)$ is given by $(y,f(x)/m)$. We can easily view several motions of a mechanical system simultaneously, and how they fit together — something that is near-impossible to do by a direct physical observation. We also looked at a second-order differential equation from electrical circuit theory that we could visualize this way.Generalization

In principle, we still have the same geometric interpretation for a system of $n$ first-order differential equations of the form

$$ \dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x}) $$

where $\boldsymbol{x}=(x_1,\ldots,x_n)$ and $\boldsymbol{f}(\boldsymbol{x})=(f_1(\boldsymbol{x}),\ldots,f_n(\boldsymbol{x}))$, where each $f_i$ is a function from $\mathbb{R}^n$ to $\mathbb{R}$. We saw an example for $n=3$, namely a three-species food chain. Of course, a direct visualization beyond $n\geq 3$ is difficult but one can look at different projections of lower dimension.It is not hard to generalize the case of one-dimensional second-order differential equations to higher dimensions. If we consider, for instance, a particle in $\mathbb{R}^3$ subject to a force $\boldsymbol{f}(\boldsymbol{x})$ depending only on its position $\boldsymbol{x}$, then we can set $y_i=\dot{x}_i$, $i=1,2,3$, and rewrite the three equations of motion as the six equations

$$ \begin{cases} \dot{x}_i=y_i\\ \dot{y}_i=\frac{f_i(\boldsymbol{x})}{m}\, ,\quad i=1,2,3. \end{cases} $$

Therefore, the motion of particle in $\mathbb{R}^3$ can be completely represented as the motion of a point in a six-dimensional phase space, namely $\mathbb{R}^6$. If we now have $N$ particles, each of them is described by six numbers at each time $t$, three for the position and three for the velocity, it can be completely described as point moving in the phase space $\mathbb{R}^{3N}$.Another generalization that the reader can guess is when we have differential equations of order higher than two. In brief, the fundamental class of differential equations is that of first order, to which one can bring a differential equation of higher order by introducing sufficiently many new coordinates to get the appropriate phase space. A point in this abstract space gives the complete description of the system at any given time $t$, and a solution of the original equation can be visualized as the motion of a particle in an abstract fluid flow in this phase space.

Possible long-term behaviors of solutions.

It is not by accident that we saw an increase of the complexity of the behavior of solutions in our examples as the dimension of their phase space increased.

In dimension one, all solutions have to behave monotonically or remain constant (fixed point).

In dimension two, there is ‘more room’ for solutions: in addition to constant solutions (equilibria), we observed periodic solutions (closed trajectories). In dimension three, we observed a phenomenon that turns out to be impossible in dimension two, namely ‘chaotic attractors’. (Of course, we leave aside solutions that escape to infinity that is a possible (uninteresting) behavior in any dimension.)

Uniqueness of solutions $=$ determinism $\neq$ predictability.

As we mentioned, we did not face the problem of existence and uniqueness of the solutions of differential equations. This will be done in detail in the sequel. It turns out that existence and uniqueness is guaranteed under a general assumption. Uniqueness has a very important consequence : *two trajectories in phase space can never intersect*. This property is called *determinism* in physics: given an initial state of system, its evolution in the future (and in the past, that is, looking backward in time) is completely determined. But *determinism is not synonymous with predictability*, as the three-species food chain example suggests. It is indeed possible that a tiny change in the initial state of the system leads to a totally different future. This is deterministic chaos.

About solving differential equations on a computer.

Since the viewpoint in this ebook is to use digital experiments, one should be concerned about what a computer actually draws. As a matter of fact, we are obliged to discretize a differential equation and therefore we are computing an approximation to the true solution, which, by the way, can seldom be written in elementary terms. This is the subject of *numerical methods* that we will not deepen in this ebook. We content ourselves by mentioning that we use standard powerful numerical methods such as the Runge-Kutta method.