The Controls

The Controls page gives you direct access to all three EOC system parameters via sliders. After adjusting parameters, click COMPUTE to recalculate the trajectory, then ANIMATE to watch it evolve. Four diagnostic panels update in parallel:

• Trajectory divergence: Two nearby trajectories evolving from almost-identical initial conditions. Watch them separate exponentially in the chaotic regime.
• Bifurcation diagram: Consolidation variable C plotted against rho across its full range, showing the classic Lorenz structure: stable fixed points, period-doubling cascade, chaos, and periodic windows.
• Lyapunov exponent landscape: The largest Lyapunov exponent λ₁ as a function of rho. Positive λ₁ marks the chaotic regime; zero marks the bifurcation boundary.
• Phase portrait: The attractor in the E-O plane.

• σ (sigma) — Engagement tracking rate: How quickly E responds to changes in O. Range: 1–20. Default: 10.
• ρ (rho) — Driving intensity: The control parameter. Below ~1: all activity decays. Below ~24.74: stable fixed points. Above ~24.74: chaotic wing transitions. Range: 0.5–50. Default: 28.
• β (beta) — Consolidation decay rate: How fast patterns erode without reinforcement. Low beta = durable change; high beta = fragile change. Range: 0.5–10. Default: 8/3 ≈ 2.667.

Toggle the second trajectory to observe sensitive dependence on initial conditions: two trajectories separated by a tiny initial distance diverge exponentially in the chaotic regime. This is the mathematical basis for why individual therapeutic outcomes within a chaotic episode are fundamentally unpredictable.

The formal framework is on The Science page.