Attractors

The time evolution of the Lorenz system is described by the following differential equation:¹

\begin{align} \begin{split} \dot{x} &= \sigma (y - x), \\ \dot{y} &= x (\rho - z) - y, \\ \dot{z} &= x y - \beta z. \end{split} \end{align}

Here \(\sigma\), \(\rho\) and \(\beta\) are real (positive) parameters.

Edward Lorenz originally studied the system for \(\sigma = 10\), \(\rho = 28\) and \(\beta = \frac{8}{3}\), see here. Keeping \(\sigma\) and \(\beta\) fixed, the system has two non-zero stable fixed points as long as

\begin{align} \rho < \sigma \frac{\sigma + \beta + 3}{\sigma - \beta - 1}. \end{align}

For Lorenz' original values of \(\sigma\) and \(\beta\) the critical point where the inequality no longer holds is at \(\rho = \frac{470}{19}\). Varying \(\rho\) slightly around the critical value, we can see how the system changes from being attracted to the fixed points to being repelled by it, see here.

Another interesting point is \(\rho = 99.96\) where the system has knotted periodic orbits, see here.

The equations for the Rössler system are

\begin{align} \begin{split} \dot{x} &= -y - z, \\ \dot{y} &= x + a y, \\ \dot{z} &= b + z (x - c), \end{split} \end{align}

with parameters \(a\), \(b\) and \(c\).

Rössler initially (see here) studied the case \(a = 0.2\), \(b = 0.2\), \(c = 5.7\) which can be seen here in the visualization above.

The Rössler attractor is full of periodic orbits and for some values of the parameters all orbits are periodic. For \(a = 0.1\), \(b = 0.1\) and \(c = 8.5\) for instance, the attractors winds around the origin in the \(x\)-\(y\)-plane four times. Here is a case where the parameter \(c\) changes between \(c = 8.5\) and \(c = 9\) every few seconds which shows the transition of the attractors between the periodic and the chaotic behaviour.

The Thomas system is a cyclically symmetric system described by the system of equations

\begin{align} \begin{split} \dot{x} &= \sin(y) - b x, \\ \dot{y} &= \sin(z) - b y, \\ \dot{z} &= \sin(x) - b z, \end{split} \end{align}

with only one parameter \(b\). The original paper also considers and plots some related systems, e.g. one where the \(\sin)\) term is replaced by polynomial.

The critical point at which the system becomes chaotic is \(b \approx 0.208\). Here is the situation just above the critical point (\(b = 0.22\)) and here just below (\(b = 0.18\)).

The modified Chua attractor (see here, section 3.1) is an instance of a multiscroll attractor given by the equations

\begin{align} \begin{split} \dot{x} &= \alpha (y - h), \\ \dot{y} &= x - y + z, \\ \dot{z} &= -\beta y, \end{split} \end{align}

where \(h\) is the following combination:

\begin{align} h = \begin{cases} \frac{b \pi}{2 a} (x + 2 a c) & x \leq -2 a c, \\ -b \sin\left(\frac{\pi x}{2 a} + d\right) & -2 a c < x < 2 a c, \\ \frac{b \pi}{2 a} (x - 2 a c) & x \geq 2 a c. \end{cases} \end{align}

The parameters of the system are \(\alpha\), \(\beta\), \(a\), \(b\), \(c\) and \(d\).

The parameter \(c\) controls the number of "wheels", see for instance here for the case \(c = 7\).

The van der Pol system is a second-order system, but it can be transformed into a first-order two-dimensional system with the following equations:

\begin{align} \begin{split} \dot{x} &= y, \\ \dot{y} &= \mu \left(1 - x^2\right) y - x. \end{split} \end{align}

The parameter is \(\mu\). For the visualization the motion is forced to the \(z = 0\) plane with the extra equation \(\dot{z} = -z\).

Here is a visualization of the transition of \(\mu\) from positive to negative values. For positive values we can see a limit cycle. As \(\mu\) decreases and passes zero, a fixed point appears at the center to which the particles spiral.