This info sheet goes along with this interactive dynamical systems demo.
Lorenz Attractor
A simple, 3-D, stripped-down weather model which gave rise to the entire field of Chaos research. Produces the famous ‘butterfly attractor’.
\[\dot{x} = \sigma(y -x)\] \[\dot{y} = x(\rho - z) - y\] \[\dot{z} = xy - \beta z\] \[\{ \sigma, \rho, \beta \} = \{10, 28, \frac{8}{3} \}\]Chen-Lee Attractor
A lesser known chaotic attractor. Produces beautiful spirals and sometimes a thin wormhole between them.
\[\dot{x} = ax - yz\] \[\dot{y} = by + xz\] \[\dot{z} = dz + \frac{1}{3}xy\] \[\{a, b, d\} = \{5, -10,-0.38 \}\]Rössler Attractor
A well-known chaotic attractor with a characteristic ‘hump’. Trajectories go around in a loop, travel over the hump, come back down onto the loop.
\[\dot{x} = -y - z\] \[\dot{y} = x + ay\] \[\dot{z} = b + z(x-c)\] \[\{a, b, c\} = \{0.2, 0.2,14 \}\]Van der Pol
Typically the first system with a ‘limit cycle’ students see. Basically a mass on a spring, where the spring has nonlinear damping. All trajectories are attracted towards a unique limit cycle.
\[\dot{x} = \mu(x - \frac{1}{3}x^3 - y)\] \[\dot{y} = \frac{x}{\mu}\] \[\dot{z} = -z\] \[\{\mu\} = \{1\}\]NOTE: This \(\dot{z} = -z\) is not part of the Van der Pol system. Van der Pol is 2-D. I include this \(z\) term to just decay trajectories onto the \(xy\) plane in the simulation.
Line Attractor
A generic line attractor. There’s a line in state space and all trajectories are sucked onto it.
\[\dot{x} = 0\] \[\dot{y} = mx + b - y\] \[\dot{z} = -z\] \[\{m,b\} = \{3,-18\}\]Multiple Point Attractors
A system with multiple (in this case infinitely many) point attractors. This is a simple extension of example 2.4.1 in Strogatz to three dimensions, instead of one.
\[\dot{x} = -\alpha\sin(\omega x)\] \[\dot{y} = -\alpha\sin(\omega y)\] \[\dot{z} = -\alpha\sin(\omega z)\] \[\{\alpha,\omega\} = \{50,\frac{1}{10}\}\]Conservative Double Well System
A Newtonian particle with unit mass moving in a double-well potential. The trajectories live on closed curves defined by contours of constant energy. This is from example 6.5.2 in Strogatz.
\[\dot{x} = y\] \[\dot{y} = x-x^3\] \[\dot{z} = -z\]NOTE: This \(\dot{z} = -z\) is not part of the system. Same as the Van der Pol, I include this \(z\) term to just decay trajectories onto the \(xy\) plane in the simulation.
Contracting Dynamical System, with Input
A contracting recurrent neural network, driven by external input. Contracting systems forget their initial conditions exponentially.
\[\dot{x} = -x + \tanh( \alpha(x - y - z) ) + \beta\cos(t)\] \[\dot{y} = -y + \tanh(\alpha(x - y - z)) + \beta\sin(\frac{t}{2} + \frac{t}{\pi})\] \[\dot{z} = -z + \tanh(\alpha(x + y + z)) + \beta\cos(t)\] \[\{\alpha,\beta\} = \{0.32258065,100\}\]Contracting Dynamical System, without Input
The same system as above, with \(\beta = 0\).