The Hénon map was introduced by the French astronomer and mathematician Michel Hénon in 1976 as a deliberately simplified model of the Poincaré section of the Lorenz system. Hénon wanted a two-dimensional map — rather than a three-variable system of ODEs — that would be far cheaper to iterate on a computer while still reproducing the essential qualitative behaviour of a strange attractor: a bounded, non-repeating trajectory with sensitive dependence on initial conditions. The map is defined by two simple equations, xn+1 = 1 − a·xn² + yn and yn+1 = b·xn, where a controls the strength of the nonlinear folding and b controls how strongly the previous x-value is fed back as the new y-value (b also sets the Jacobian determinant, −b, i.e. the rate of area contraction per iteration). At the classic parameters a = 1.4 and b = 0.3 — the values Hénon himself studied — the map settles onto a strange attractor rather than diverging to infinity or collapsing to a periodic cycle, and it has since become one of the most widely studied toy models in chaos theory precisely because its two equations are so easy to iterate yet so rich in structure.
Under repeated iteration the classic Hénon attractor reveals a fractal structure: a cross-section through the attractor looks like a Cantor set, an uncountable dust of points with gaps at every scale, and the whole attractor is self-similar — zooming into any dense filament reveals a near-copy of the larger structure, layer after layer. Its box-counting (capacity) dimension has been measured at approximately 1.26, meaning it is more than a simple curve (dimension 1) but does not fill an area (dimension 2) — the signature of a genuine fractal. This simulation iterates the map directly, with no differential-equation solver needed since it is already a discrete-time system, and accumulates tens of thousands of points on the canvas without clearing the frame, so the point cloud visibly fills in and the density of the attractor becomes apparent. Use the magnification slider to zoom into the attractor and watch the self-similar filament structure repeat at finer and finer scales, and use the a-slider to explore periodic windows — narrow ranges of a, such as around a ≈ 1.22–1.25, where the chaotic attractor collapses into a simple stable periodic orbit before chaos resumes as a is increased further.
What is the Hénon map and why was it invented?
The Hénon map is a discrete two-dimensional iteration, xn+1 = 1 − a·xn² + yn, yn+1 = b·xn, published by Michel Hénon in 1976. He designed it as a simplified stand-in for the Poincaré section of the Lorenz system: the Lorenz equations require integrating three coupled ODEs to see a strange attractor, whereas the Hénon map produces one from a handful of arithmetic operations per step, making it vastly cheaper to study on the computers of the 1970s while preserving the same qualitative chaotic behaviour.
What makes the Hénon map chaotic?
The xn² term folds the plane nonlinearly, stretching distances apart in one direction while the b·xn feedback and the overall contraction (the map contracts area by a factor of b on every iteration) fold the stretched sheet back on itself, layer after layer, like a baker kneading dough. This repeated stretch-and-fold action is exactly the mechanism that produces sensitive dependence on initial conditions and a positive Lyapunov exponent — two initial points that start extremely close together separate exponentially after only a few dozen iterations.
What is the strange attractor's fractal dimension?
At the classic parameters (a = 1.4, b = 0.3) the Hénon attractor has a box-counting (capacity) dimension of approximately 1.26 — a non-integer value that places it strictly between a one-dimensional curve and a two-dimensional area. A cross-section perpendicular to the attractor's main filament looks like a Cantor set: an infinitely fine dust of points with gaps at every level of magnification, which is the hallmark of a fractal strange attractor rather than an ordinary smooth curve.
As a is decreased from 1.4 the attractor typically passes through a sequence of period-halving bifurcations, punctuated by narrow periodic windows where the chaotic cloud collapses onto a small number of discrete points (a stable periodic orbit) instead of filling in a fractal cloud. A well-known periodic window sits around a ≈ 1.22–1.25 at b = 0.3. The parameter b sets the rate of area contraction (the map's Jacobian determinant is constant and equal to −b); larger b values thin the attractor's filaments less aggressively, while very small b values make the map close to one-dimensional (similar to the logistic map) and very large b values can cause orbits to diverge to infinity instead of settling onto an attractor.
Because the Hénon attractor is built from infinitely many stretch-and-fold operations layered on top of each other, any small patch of the attractor, when magnified, looks statistically similar to the whole attractor — the same banana-shaped arcs and gaps reappear at every scale, a property called self-similarity. This simulation's zoom control rescales the view around the attractor's centre without erasing the accumulated points, so you can watch a single thick-looking filament resolve into several thinner parallel filaments, and those filaments resolve into even more filaments, as you increase the magnification.
A Poincaré section takes a continuous three-dimensional flow, like the Lorenz system, and records the points where the trajectory crosses a fixed plane, turning a continuous-time attractor into a discrete-time map on that plane. Hénon studied the qualitative shape of such sections of the Lorenz attractor and then constructed his own map from scratch to have similar folding behaviour while being vastly simpler to compute — the Hénon map is not derived algebraically from the Lorenz equations, but it captures the same essential stretch-and-fold mechanism believed to underlie many real strange attractors.
If you start two iterations from points that differ by only a tiny amount, e.g. 10⁻¹⁰, the nonlinear folding of the Hénon map amplifies that difference roughly exponentially with each iteration once the orbit is on the chaotic attractor. After a few dozen iterations, the two trajectories have completely decorrelated even though they started almost identically — the same phenomenon popularly known as the butterfly effect, first identified in continuous systems like the Lorenz equations but present here in a much simpler, purely algebraic map.
The Hénon map does not have a global attractor for every choice of a and b — for sufficiently large a, or for initial points far from the basin of attraction, iterates can grow without bound rather than settling onto the fractal set. This simulation restricts the sliders to the ranges (a between 0.9 and 1.4, b between 0.1 and 0.4) in which, starting from the origin, the map reliably converges onto a bounded attractor or periodic orbit, so you can explore the interesting dynamics without the trajectory escaping to infinity.
"Classic (a=1.4, b=0.3)" reproduces Hénon's original 1976 parameters and the canonical strange attractor with dimension ≈ 1.26. "Periodic window (a=1.25, b=0.3)" sits inside one of the narrow windows of stability where the chaotic behaviour temporarily disappears and the orbit settles onto a small periodic cycle instead of filling in a fractal cloud. "a=1.3 (near period-doubling)" sits in a transitional region between the periodic window and the fully chaotic regime, where the attractor's structure is still forming and period-doubling-like behaviour can be seen as a is tuned finely.