The Lorenz attractor at ρ = 28

• the map has an orbit of least period 1 ×6
• above the critical parameter the trajectory visits both lobes ×1
• and at the finest it is a few per cent ×1
• and each is narrower by the stretch factor ×1
• and even at the cusp, where a straight line cannot fit, the departure stays small ×1
• and it is the Lorenz system's own exponent, near nine tenths ×1
• and never rises above it again ×1
• and no window it finds contains 1.4 ×1
• and settles on neither fixed point ×1
• and the strips are laid out in order without overlapping ×1
• and there are parameters nearby where it is negative — periodic windows ×1
• areas shrink at every step ×1
• at a = 1.4 the exponent is about 0.42 ×1
• at every finer grid, 90% of the time fits in a smaller share of the area ×1
• below the critical parameter it spirals into a fixed point ×1
• between two and four starting gaps, each a whole power of ten from −2 to −12 ×1
• both orbits are certified to exist and to attract ×1
• each cycle returns to its start ×1
• each image has 0.3 times the area of the one before ×1
• each run passes many peaks ×1
• enough returns to see a shape ×1
• every box's interval image lies inside the set of boxes ×1
• every parameter sampled in the proved region has a positive exponent ×1
• every periodic point found is a saddle ×1
• every point of the boundary lands strictly inside the quadrilateral ×1
• every surviving strip lies inside the square ×1
• every window still holds some of the orbit ×1
• half the time is spent in under a quarter of the cells the orbit visits ×1
• Krawczyk's test certifies a stable periodic orbit ×1
• Newton's method converges to the orbit ×1
• no box's image leaves the grid ×1
• no periodic orbit's average is the typical one ×1
• orbits from different starts spend their time the same way ×1
• scans close to 1.4 at ever finer spacing find fewer windows, and the finest finds none ×1
• so the total width falls, and the limit has none ×1
• the closed-form fixed points really are fixed ×1
• the contraction is between 0.1 and 0.4 ×1
• the direction never leaves the cone and every step stretches by at least 1.4 ×1
• the first image lies inside the region, so every image lies inside the one before ×1
• the fixed point maps to itself ×1
• the geometric parameter is between 0.1 and 8 ×1
• the growing set of boxes closes up ×1
• the growth rate does not depend on how close the two starts were ×1
• the Hénon parameter is between 1 and 1.45 ×1
• the Jacobian determinant is the contraction ×1
• the log of the volume falls at the rate σ + 1 + β ×1
• the long averages agree ×1
• the map's maximum is in its interior, so it folds ×1
• the nearest window to 1.4 the scan finds has period 19, at 1.40016 ×1
• the number of magnifications is a whole number between 1 and 4 ×1
• the number of stages is a whole number between 2 and 5 ×1
• the number of steps is a whole number between 500 and 40000 ×1
• the option is one this view reads ×1
• the orbit at the fixed point averages something else ×1
• the orbit stays bounded, which is what makes it an attractor ×1
• the Prandtl parameter is between 0.1 and 40 ×1
• the Rayleigh parameter is between 0.1 and 100 ×1
• the return map is a curve, not a cloud — its typical thickness is under one per cent of its range ×1
• the scan finds the stable orbit at the stated parameter ×1
• the step size is between 0.0005 and 0.05 ×1
• the stretching factor is between 2.2 and 5 ×1
• the strips double at every stage ×1
• the surviving set has a dimension between nought and one ×1
• the trace runs from +1 at the saddle-node to −1 at the period doubling ×1
• the trajectory gets below the bound ×1
• the trajectory stays in a bounded region ×1
• the two starts give the same distribution of peaks ×1
• the windows of period ten or less are typically more than ten times wider than those of period nineteen or more ×1
• wherever V is above the ellipsoid's bound it is falling ×1

The whole library · What the figures prove