Lyapunov Spectrum via Benettin QR

The Henon map x_n+1 = 1 − a x_n² + y_n, y_n+1 = b x_n is the canonical 2D chaos benchmark. At (a, b) = (1.4, 0.3) its attractor is a strange set with a positive largest Lyapunov exponent and a negative second exponent that sum exactly to ln|b|. Drag the handle on the parameter panel to explore how the attractor morphs and how the spectrum responds. The trace identity λ₁ + λ₂ = ln|b| is an exact dynamical invariant that the Benettin QR algorithm preserves to machine precision.

Figure 1. Henon attractor (left) and the (a, b) parameter panel (right). Method: 2D Henon iteration plus a Benettin QR algorithm on a 2x2 tangent frame with modified Gram-Schmidt re-orthonormalization at each step.

WHAT TO TRY

Drag the (a, b) handle across the Henon parameter plane: the attractor morphs and the two Lyapunov exponents shift. Wherever lambda_1 crosses above zero (the shaded bands) the motion is chaotic.
Read the constraint lambda_1 + lambda_2 = ln|b|: the map contracts area at a fixed rate, so the two exponents always sum to the same value. A positive lambda_1 forces a more negative lambda_2.
Park on a periodic window (an unshaded gap): both exponents go negative and the strange attractor collapses to a few clean points, the order hiding inside the chaos.