# Graphics Bifurcation

Lorenz ‘96, a set of ordinary differential equations with extraordinary behaviour, switching from decay, to periodicity, to chaos and all the way back again, according to the setting of a single parameter.

The equations were introduced by (and named for) Edward Lorenz in studying the dynamics of the Earth’s atmosphere. Lorenz was one of the pioneers of chaos theory, coining the term butterfly effect after investigating an earlier model known as the strange attractor. Chaotic behaviour is characterised by sensitivity to initial conditions: small changes in the starting state of a chaotic system can cause dramatic changes in its future states.

If $x$ is a $D$-dimensional vector and $F$ the parameter, the Lorenz ‘96 equations are given by:$\frac{\mathrm{d}x_{d}}{\mathrm{d}t} = x_{d-1}(x_{d+1}-x_{d-2})-x_{d}+F,$ with subscript indices interpreted cyclically.

These particular images reveal the bifurcation of the 8-dimensional (i.e. $D = 8$) Lorenz ‘96 equations as $F$ increases from left to right. The first image is approximately on the interval $F \in [0,7]$, the second two zooming in towards the region of $F \approx 5$. Smoother vertical bands suggest chaotic behaviour, while periodicity manifests as tendril-like structures.

## References

Lorenz, E. N. (2006). Predictability—A Problem Partly Solved, chapter 3 of Predictability of Weather and Climate. Cambridge University Press, pp40–58.