Part III: Beyond flows in the plane: quasi-periodicity & chaos - Chapter 3

Four more strange attractors

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:

$$ \begin{cases} \dot{x}=x\,(1-x)-f_1(x)\,y\\ \dot{y}=f_1(x)\,y - f_2(y)\,z-d_1 y\\ \dot{z} = f_2(y)\,z-d_2 z \end{cases} $$


$$ f_1(x)=\frac{a_1 x }{1+b_1 x}\quad\text{and}\quad f_2(y)=\frac{a_2 y }{1+b_2 y}. $$

The species with density $x$ is at the lowest level of the chain. It is consumed by the intermediate species with density $y$ which is itself consumed by the species with density $z$ that is at the top of the chain and has no predator. In the absence of the top predator, the system boils down to

$$ \begin{cases} \dot{x}=x\,(1-x)-f_1(x)\,y\\ \dot{y}=f_1(x)\, y-d_1 y. \end{cases} $$

You can recognize the prey-predator model of Rosenzweig-McArthur. In the digital experiment we take

$$ a_1=5,\thinspace a_2=0.1, \thinspace b_2=2, \thinspace d_1=0.4,\thinspace d_2=0.01,\thinspace \text{and}\thinspace b_1\in [2,6.2]. $$

You can observe a chaotic strange attractor for some values of $b_1$.

An attractor with spiral structure

We show a strange attractor proposed by Arneodo, Coullet and Tresser in 1981. It is generated by the equations

$$ \begin{cases} \dot{x}=y\\ \dot{y}=z\\ \dot{z} = ax-by-z-cx^3 \end{cases} $$

with $a=5.5$, $b=3.5$, and $c=1$.

Chua’s circuit and the double-scroll attractor

In the mid-1980s, Chua modeled a circuit that was a simple oscillator exhibiting a variety of bifurcation and chaotic phenomena. In dimensionless form, the equations are

$$ \begin{cases} \dot{x}=a\, (y-x-g(x))\\ \dot{y}=x-y+z\\ \dot{z} = -by \end{cases} $$

where $a,b$, and $c$ are dimensionless parameters. The function $g(x)$ has the form

$$ g(x)=cx+\frac{1}{2} (d-c)\big(|x+1|-|x-1|\big) $$

where $c$ and $d$ are constants. Before nondimensionalization, $x$ and $y$ represent a voltage, and $z$ a current.
Here we just show the so-called double-scroll attractor which shows up for the values

$$ a=15, \thinspace b=25.58,\thinspace c=-5/7, \thinspace d=-8/7. $$

Rabinovich-Fabrikant equations

In 1979, Rabinovich and Fabrikant proposed the equations

$$ \begin{cases} \dot{x}=y\, (z-1+x^2)+\gamma x\\ \dot{y}=x\, (3z+1-x^2)+\gamma y\\ \dot{z} = -2z\,(\alpha+xy) \end{cases} $$

where $\alpha,\gamma$ are parameters. It was obtained after the simplification of a model of self-modulation of waves in nonequilibrium media. When $\gamma=0.87$, $\alpha=1.1$, there is a strange attractor.