Concept

Renormalisation

Rescaling a small part of a system so that it looks like the whole, then studying the rescaled system instead. In dynamics it explains why small copies of a picture reappear inside it, and why some constants are the same across many different systems.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them.

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

dynamics · Period-doubling

Where the free critical point goes, for every cubic in a family. The parameter plane of the cubics (z − 1)(z² + z + a) from −2 to 3 across and −2.5i to 2.5i up, coloured by the root Newton's method takes 0 to, with small dark regions where 0 is trapped in a cycle.

Small Mandelbrot sets among the cubics

For most cubics Newton's method finds a root from almost every start. For some it does not: a whole region of starting points falls into a cycle and never finishes. Which cubics are the bad ones is decided by a single starting point — the one critical point that is not a root — and when the cubics are laid out in a plane and coloured by where that point goes, the bad ones form small, exact copies of the Mandelbrot set.

dynamics · Newton basins

Named alongside it

The objects these essays reach for when they reach for this one.

Period-doubling Basin of attraction Bifurcation Convergence Critical point Feigenbaum constant Logistic map Mandelbrot set Newtons method Periodic orbit Scaling Self-similarity

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