Aizawa Attractor

The system

The Aizawa system is usually written with its axis along z. Wavelace is y up, so here y and z are swapped: y is the height and the axis is vertical. Its usual constants are written in as numbers, a = 0.95, b = 0.7, c = 0.6, d = 3.5 and e = 0.25. The last, f, is on a slider at 0.1.

The x and z expressions turn every point about the vertical axis at 3.5 radians a second, a turn every 1.8 seconds at Speed 1. They also move it out from the axis above the height 0.7 and in toward it below.

The height expression holds the rest. On the axis it is the cubic 0.6 + 0.95y − y³/3, whose top root, y ≈ 1.94, draws the height up. The term in x² + z² pulls it down again away from the axis.

The shape

Together these make a loop. A speck rises up the axis inside a narrow tube, is pushed out near the top where the height is past 0.7, and falls down the outside of a round shell. Below 0.7 it is drawn back in and rises again. The shell reaches 1.5 from the axis, and the height runs from −0.37 to 1.85.

Why it never repeats

At f = 0 the system looks the same from every side of the axis. Then only the distance from the axis and the height matter, a system of two. The Poincaré–Bendixson theorem says a system of two cannot be chaotic. The loop settles into a cycle every 6.05 seconds, and the turning spreads it over a torus.

The term f y x³ breaks that symmetry and lets the third direction in. At the opening f = 0.1 two specks started 10⁻⁸ apart grow apart by about e^(0.11t), the size of the whole attractor after 150 to 200 seconds.

Try

• Drag f to 0. The chaos goes, and the specks wind round a smooth torus.
• Set Seeds to 1. One speck, its trail showing the loop: up the tube, out over the top, down the shell.
• Press Top in the deck. The turning about the axis, and the tube down its middle.
• Raise Trail to draw more of each path, and the shell fills in.

Read more

• Attractor
• Chaos theory
• Poincaré–Bendixson theorem
• Lyapunov exponent