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Concept

Dissipation

The contraction of volume by a map or flow, so that a region of starting points shrinks as it is carried along. It is what forces an attractor to have no volume, and its rate is the sum of the Lyapunov exponents, which is the logarithm of the determinant of the derivative.

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

The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates.

Two lobes and no cycle

Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

dynamics · Strange attractor A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box.

A dimension from the stretching rates

An attractor has no construction rule, so its dimension has to be counted — that was the rung below's argument for defining dimension by counting at all. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.

dynamics · Fractal dimension

Named alongside it

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

AttractorSensitive dependenceBifurcationBox dimensionFixed pointFractal dimensionGram schmidtIterated function systemLorenz systemLyapunov exponentOrbitScaling

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