Nonlinear Dynamics Laboratory
Deterministic, yet unpredictable.
The Lorenz system is fully determined by three equations — and still, two almost identical starting points end up on completely different paths. This is the dynamical heartbeat attached to every vertex of the manifold.
Governing equations
dy/dt = x(ρ − z) − y
dz/dt = xy − βz
Documented baseline
| σ (Prandtl-like) | 10 |
| ρ (Rayleigh-like) | 28 |
| β | 8/3 ≈ 2.667 |
| Integration step Δt | 0.005 |
| Inter-vertex coupling ε | 0.12 |
Parameters
Parameter changes restart the trajectories and the Lyapunov estimate so every number corresponds to the current settings.
Time series · x(t)
Separation · log₁₀ ‖A − B‖
Roughly linear growth on this log scale is exponential divergence; it saturates once the separation reaches the size of the attractor.
Largest Lyapunov exponent
warming up…
Method. Benettin two-trajectory renormalisation: a reference and a companion orbit start 10⁻⁸ apart (after a transient); every 10 steps the log-growth of their distance is accumulated and the companion is rescaled back to 10⁻⁸. λ₁ ≈ Σ ln(dₖ/d₀) / T.
Estimate converges slowly and depends on Δt and integration time. For the classical parameters the literature commonly reports λ₁ ≈ 0.9. Positive λ₁ indicates chaos; negative values indicate convergence to a fixed point (try ρ < 24).
Coupled-oscillator synchronization
Eight identical Lorenz oscillators on a ring, each coupled to its two neighbours by ε Σ(sⱼ − sᵢ) on all three variables, started from random states.
Sync error = √(mean over ring edges of ‖sᵢ − sⱼ‖²), plotted as log₁₀. For identical-variable (full-state) coupling, the master stability function is λ₁ − α, so the synchronized state is linearly stable when ε·λ₂ > λ₁ (using the live λ₁ estimate). The documented mesh coupling ε = 0.12 is far below that threshold on this ring.
Three ideas worth keeping
Deterministic chaos
Technical. A bounded, aperiodic solution of a deterministic system with at least one positive Lyapunov exponent; trajectories live on a strange attractor of fractal dimension ≈ 2.06.
Plainly. No dice are rolled, yet the motion never repeats and can't be forecast far ahead.
Sensitivity to initial conditions
Technical. Infinitesimal perturbations grow on average as e^{λ₁t}; prediction horizon scales with ln(1/δ)/λ₁.
Plainly. Making your starting measurement 1000× more precise only buys a few extra time units of accurate prediction.
Nonlinear coupling
Technical. Diffusive coupling over the mesh graph pulls neighbouring states together; whether they synchronize depends on ε, the graph Laplacian spectrum and the coupling scheme.
Plainly. Neighbours influence each other — and the mesh wiring decides who counts as a neighbour. That's why rewiring the mesh can matter.