Henon Map: A Strange Attractor from a Simple Iteration

A quadratic map with just two parameters

In 1976 the astronomer Michel Henon was looking for the simplest possible system that would reproduce the strange attractor Edward Lorenz had found in his continuous-time weather model, but in a discrete map that would be far cheaper to iterate and easier to analyse. He arrived at:

x_next = 1 - a*x^2 + y
y_next = b*x

With the classic parameter values a = 1.4 and b = 0.3, iterating this pair of equations from almost any starting point produces, after the first few transient steps, an intricate curved band that never closes on itself and never repeats -- the Henon attractor. The map is invertible (you can solve for the previous point from the current one), area-contracting rather than area-preserving, and its Jacobian determinant is the constant -b, which is why the attractor has zero area even though it lives in a two-dimensional plane.

Stretch, fold, repeat

The quadratic term -a*x^2 does the stretching: nearby points with different x values get pushed apart by different amounts, growing the separation between initially close orbits exponentially -- the hallmark positive Lyapunov exponent of a chaotic system. The linear term b*x then folds that stretched strip back into a bounded region on the next iterate. Repeat that stretch-fold cycle thousands of times and the strip gets folded over itself so many times that, zoomed in, you find the same folded structure again at a smaller scale: the attractor has a fractal (non-integer) dimension, measured for the classic parameters at approximately 1.26 -- more than a line, less than a filled area.

Sensitive dependence, made visible

Because the map stretches distances along one direction and contracts them along another, two points that start a millionth of a unit apart diverge to completely different-looking trajectories within a few dozen iterations, even though both eventually trace out the same overall attractor shape. This is the practical meaning of deterministic chaos: the rule is completely deterministic (no randomness anywhere in the equations) and yet long-term prediction of any individual orbit is impossible without infinite-precision knowledge of the starting point, because rounding error itself grows exponentially under iteration.

Why this map still matters

The Henon map became the standard toy model for strange attractors precisely because it is cheap to compute (two multiplications, two additions per step) yet exhibits essentially all the qualitative features of much more expensive continuous flows: a fractal invariant set, positive Lyapunov exponents, and a mix of stable and unstable periodic orbits densely embedded inside the chaotic attractor. It remains a standard testbed for numerical techniques -- estimating fractal dimension, computing Lyapunov exponents, finding periodic orbits -- precisely because its exact, closed-form iteration rule lets researchers check a numerical method's answer against ground truth.

Frequently asked questions

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Everything above runs in your browser — open Henon Map and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.