Generative eBooks

From quasiperiodicity to chaos

Generative eBooks — 2016-06-08T13:04:08Z

Apart from fixed points, closed trajectories and strange attractors, there is a fourth major type of attractor, namely a torus (generalization of a circle) on which solutions may wrap densely. This phenomenon is called quasiperiodicity. We shall start with the simplest example of quasiperiodicity before studying the planar double pendulum which exhibits periodicity, quasiperiodicity and chaos.
A toy example: a pair of undamped harmonic oscillators
At the beginning of this ebook, we (…)

- Part III: Beyond flows in the plane: quasi-periodicity & chaos

Four more strange attractors

Generative eBooks — 2016-06-08T13:03:31Z

In this section, you can observe attractors arising in different domains. Our goal is to show chaotic strange attractors with different shapes arising in diverse contexts.
A chaotic three-species food chain
In 1991, Hastings and Powell proposed the following model of a three-level food chain: $$ \begincases \dotx=x\,(1-x)-f_1(x)\,y\ \doty=f_1(x)\,y - f_2(y)\,z-d_1 y\ \dotz = f_2(y)\,z-d_2 z \endcases $$ with $$ f_1(x)=\fraca_1 x 1+b_1 x\quad\textand\quad f_2(y)=\fraca_2 y 1+b_2 y. $$ The (…)

- Part III: Beyond flows in the plane: quasi-periodicity & chaos

The Rössler attractor

Generative eBooks — 2016-06-08T13:02:38Z

In 1976, Rössler constructed the following three-dimensional system $$ \begincases \dotx=-y-z\ \doty=x+a y\ \dotz = b+z(x-c) \endcases $$ where $a,b$, and $c$ are parameters. Rössler was looking for a system behaving like the Lorenz system, but easier to analyze. Note that the only nonlinear term appears in the $\dotz$ equation and is quadratic. Note also that, if $z=0$, we get a linear system in the $xy$-plane. As the parameters vary, this simple system can display a wide range of behavior. (…)

- Part III: Beyond flows in the plane: quasi-periodicity & chaos

The Lorenz attractor

Generative eBooks — 2016-06-08T13:01:52Z

Introduction
We study three-dimensional systems of the form $$ \begincases \dotx=f(x,y,z)\ \doty=g(x,y,z)\ \dotz=h(x,y,z) \endcases $$ where $f,g,h:\mathbbR^3\to\mathbbR$ are continuously differentiable functions. Given an initial condition $(x_0,y_0,z_0)$, there exists a unique solution $(x(t),y(t),z(t))$ passing through $(x_0,y_0,z_0)$ at time $t=0$. Like two-dimensional systems, three-dimensional systems can have fixed points that can be locally studied by linearization. They can also (…)

- Part III: Beyond flows in the plane: quasi-periodicity & chaos