Poincaré Sections: Mapping Chaos onto a Plane

A continuous trajectory in phase space is hard to visualise and classify. Henri Poincaré's insight was to stroboscopically sample the trajectory — every time it crosses a chosen surface, record the crossing point. The resulting scatter of points, the Poincaré section, collapses a multi-dimensional flow into a 2-D picture that immediately reveals whether the dynamics is periodic, quasi-periodic, or chaotic.

1. Definition and Construction

Let a continuous-time system evolve on a phase space ℝⁿ. Choose a Poincaré surface of section Σ — a codimension-1 manifold transverse to the flow — and a sign convention for the crossing direction.

The map P encodes one full return. Studying its fixed points, stability, and iteration patterns reveals the full geometry of the continuous flow.

Practical computation

// RK4 integrator with event detection (zero-crossing)
function integrateUntilCrossing(state, f, gFn, dt) {
  let prev = gFn(state), s = { ...state };
  for (let iter = 0; iter < 1e6; iter++) {
    s = rk4Step(s, f, dt);
    const curr = gFn(s);
    if (prev < 0 && curr >= 0) {
      // bisection to refine crossing time
      return refine(s_prev, s, f, gFn, dt);
    }
    prev = curr;
  }
}

2. The Poincaré Return Map and Fixed Points

A period-T orbit of the flow corresponds to a fixed point x* = P(x*) of the map. A period-n orbit (n-cycle of the flow divided by T) maps to a period-n orbit of P. Stability is determined by the monodromy matrix M = DP(x*):

3. KAM Tori and Invariant Curves

For near-integrable Hamiltonian systems, the Kolmogorov-Arnold-Moser (KAM) theorem guarantees that most invariant tori survive small perturbations. On the Poincaré section, intact KAM tori appear as smooth closed curves. The rotation number ρ = ω₁/ω₂ characterises each torus:

At the boundary between stability and chaos the tori break up into cantori — Cantor-set remnants that act as partial barriers to transport.

4. The Standard Map (Chirikov-Taylor)

The standard map is the canonical area-preserving 2-D map that models kicked rotators, plasma confinement, and comet dynamics:

5. Period-Doubling Islands and the Birkhoff Chain

Near a rational torus ρ = p/q, perturbation theory predicts a Birkhoff island chain: p elliptic (stable) islands surrounded by p hyperbolic (unstable) saddle points. Each island can itself contain sub-islands at higher-rational rotation numbers — a self-similar hierarchy:

The width of the resonance zone scales as ΔI ~ K^{1/2}, while the gap between adjacent resonances shrinks causing resonance overlap — the Chirikov criterion for the onset of global chaos: K_c ≈ 1 when resonance half-widths overlap.

6. Poincaré Sections of the Hénon-Heiles System

The Hénon-Heiles Hamiltonian is a standard test for chaos in 2-DoF conservative systems:

7. Birkhoff-Smale Horseshoe and Homoclinic Tangles

The origin of chaos near a homoclinic point — where the unstable manifold W^u of a saddle intersects its own stable manifold W^s — was understood by Poincaré himself and formalised by Smale. Each transverse intersection implies infinitely many others, producing the horseshoe map geometry:

The "homoclinic tangle" is the fractal web of intersecting manifolds visible near separatrices in any Poincaré section of a chaotic Hamiltonian system.

8. Interactive: Standard Map Explorer

Click the canvas to add an initial condition. Each click launches a trajectory that iterates for 5 000 steps, plotting its Poincaré section points. Adjust the stochasticity K to reveal KAM tori (low K), island chains (moderate K), or global chaos (high K). Use Clear to reset.

Each colour corresponds to a different initial condition. Smooth closed curves → KAM tori; dot chains → rational resonances; scattered dots → chaos.