Generative eBooks

What we have learnt so far

Generative eBooks — 2016-06-08T12:48:39Z

We summarize the salient points of this chapter.

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

- A first tour through examples

Chaotic attractor in an ‘ecosystem' with two competing species eaten by a third one

Generative eBooks — 2016-06-08T12:47:41Z

Let us return to ecology to put forward an amazing phenomenon, namely deterministic chaos. We will see other examples later on. Our goal is not to fully analyse this phenomenon here, but rather to show how complex the behavior of an innocent-looking model can be.
We model the interaction between a predator and two competing prey populations in the simplest way. Consider first the two prey populations in the absence of the predator. If $x$ and $y$ denote their densities, then the rates of (…)

- A first tour through examples

Van der Pol's periodic attractor

Generative eBooks — 2016-06-08T12:47:03Z

The equation of Van der Pol was designed as a model for an electronic oscillator in the 1920s. We do not go into the derivation of this model. In any case it is a mathematical description of obsolete technology based on radio tubes, the predecessors of our present transistors. The equation is $$ \ddotu+\epsilon(1-u^2)\dotu+u=0 $$ where $u$ represent the voltage.

As we've done for the harmonic oscillator and the pendulum, we pass to the phase plane representation by setting $x=u$ and (...)

- A first tour through examples

A new look at the pendulum

2016-06-08T12:45:40Z

Consider the motion of the following idealized pendulum: a bob of mass $m$ is attached to one end of a massless rigid rod. The other end of the rod is pivoted so that the mass may swing in a vertical plane. We neglect both the friction of the pivot and air drag.
The swinging of the pendulum is governed by $$ \ddot\theta=-\fracgL \sin \theta $$ where $\theta$ is the angle between the rod and the downward vertical. This equation is derived in all textbooks of classical mechanics.
Phase (…)

- A first tour through examples

The harmonic oscillator

2016-06-08T12:45:03Z

We know turn to an elementary example coming from classical mechanics.
The harmonic oscillator is the simplest example of a particle of mass $m$ constrained to a straight line, which we take to be the $x$-axis, subject to a force $f(x)$ when it is located at $x$. Assuming that this force is independent of the velocity $\dotx$ amounts to excluding friction. According to Newton's second law, we have $$ m \ddotx=f(x). $$ In the case of the harmonic oscillator, $f(x)=-kx$, where (…)

- A first tour through examples

Sharks and sardines

2016-06-08T12:41:45Z

The model we are going to study serves as an introduction to the qualitative analysis of differential equations in the plane. We shall introduce the concept of phase portrait.
The model
The mathematician Vito Volterra proposed in the 1920s a model describing the interaction between sharks and sardines after he was approached by the marine biologist D'Anconna to understand some real fishing data from Adriatic Sea.
Let $x(t)$ be the density of sardines as a function of time, and $y(t)$ (…)

- A first tour through examples

The logistic equation

2016-06-08T12:37:19Z

In the short run, the equation $\dotx=ax$ with $a>0$ makes sense to describe our population of bacteria. Indeed, we can imagine that, during a short interval of time, the bacteria do not crowd each other and the population size is $e^atx_0$ if there are $x_0$ bacteria at time $0$. But as time goes on, exponential growth would lead to an absurdly large number of bacteria — exceeding even the number of atoms in the universe if taken literally!
To eliminate this major drawback, let us assume (…)

- A first tour through examples

The simplest example

2016-06-08T12:36:04Z

The simplest differential equation is $$\dotx=ax$$ where $x\in\mathbbR$ and $a$ is a real-valued parameter. Despite its simplicity, this equation allows us to introduce several important concepts for the rest of this ebook. We can interpret it as a naive model for the growth of bacterial populations.
Bacteria. Consider a container filled with a nutritive solution and bacteria. As time progresses, the bacteria reproduce (by division) and die. Let $b$ (for birth) be the rate at which the (…)

- A first tour through examples