Mira Fractal

Pick a point $(x, y)$. It jumps to a new point using this rule, and you plot every landing:¹

$x' = b\,y + F(x), \qquad y' = -x + F(x'), \qquad F(x) = a x + \frac{2(1 - a)x^2}{1 + x^2}$

After thousands of landings, a flower appears, and the number $a$ changes how many petals you get. Places to explore has examples.

The number $b$ decides what happens over time. Take any small patch of points. One jump moves and distorts the patch, but its area comes out multiplied by exactly $b$, whatever the point and whatever $a$ is.²

So with $b = 1$, no area is ever lost. With $b = 0.9998$, one jump barely changes anything, but after about 3,500 jumps the patch has half its area, and after 100,000 jumps it has almost none.

That changes the picture you get. Start at $(12.1, 0)$ with $a = 0.7$. With $b = 1$, the point keeps reaching about 15 from the center for as long as we ran it. With $b = 0.9998$, it reaches 15 only during its first few thousand jumps. After 10,000 jumps it stays within about 5, and after 20,000 within about 4.³

So the long arms come from the early part of the run, before the point has settled in. MathWorld's pictures with $b = 0.9998$ stop after 10,000 jumps, which is why they show those arms. Plot longer and they fade next to the middle, where new landings keep piling up.

The rule's two-line form is one that Igor Gumowski, at CERN, and Christian Mira, at Toulouse, used to study particle motion in accelerators.⁴

Further exploring

• Lorenz attractor, another picture drawn by following one route

Sources and notes

1. Mira Fractal. ↩
2. Calculation. The four partial derivatives of the rule are $F'(x)$ and $b$ in the first row, and $F'(x')F'(x) - 1$ and $bF'(x')$ in the second. Their determinant is $b$, so the bends in $F$ cancel out. ↩
3. Calculation. Measured on this page's orbit, up to 2,000,000 jumps. Where a falling point ends up is sensitive to tiny differences, so these distances describe this run. ↩
4. Stochastic Effects in Longitudinal Phase Space uses the same two-line recurrence with a sine in place of this $F$. Boundaries of Stochasticity Domains in Hamiltonian Systems gives the affiliations. ↩