Trajectory A
Trajectory B (perturbed)
Poincaré section — plane y = 0, dy/dt < 0 (x vs z at crossing)0 crossings
This is a 2D companion to the site's 3D Rössler attractor. Instead of a rotatable 3D scene, it integrates the exact same three differential equations — dx/dt = −y − z, dy/dt = x + ay, dz/dt = b + z(x − c) — with real 4th-order Runge–Kutta stepping and draws two flat, information-dense views: the x–y projection of the trajectory, which shows the Rössler system's characteristic single spiral that occasionally kicks up in z and jumps back inward, and a genuine Poincaré section — the actual set of points where the numerically integrated trajectory crosses the plane y = 0 (found by interpolating the real crossing between two RK4 steps, not synthesized). Sliding c from about 2.5 up through 3.5, 4.2 and 5.7 walks the system through a real period-doubling cascade into chaos, visible directly as the Poincaré section grows from one point, to two, to four, to a dense chaotic scatter. A second trajectory started just 0.00001 away in x runs alongside the first so the exponential divergence that gives chaos theory its name can be read as a number, not just seen as a visual fan-out.