Lorenz Attractor

In 1963, Edward Lorenz discovered that a simplified model of atmospheric convection produces strikingly complex, non-repeating trajectories — the first famous example of a strange attractor.

The Lorenz System

$\frac{dx}{dt} = \sigma(y - x)$ $\frac{dy}{dt} = x(\rho - z) - y$ $\frac{dz}{dt} = xy - \beta z$

Classic parameters: $\sigma = 10$, $\rho = 28$, $\beta = 8/3$

The state vector $(x, y, z)$ evolves continuously. With these parameters the system is chaotic, with a Lyapunov exponent of approximately $0.9$.

Numerical Integration: RK4

We integrate using the fourth-order Runge-Kutta (RK4) method with time step $dt = 0.01$:

$k_1 = f(s) \quad k_2 = f\!\left(s + \tfrac{dt}{2}k_1\right) \quad k_3 = f\!\left(s + \tfrac{dt}{2}k_2\right) \quad k_4 = f(s + dt\,k_3)$

$s_{n+1} = s_n + \frac{dt}{6}(k_1 + 2k_2 + 2k_3 + k_4)$

Key Properties

• Sensitivity to initial conditions: two trajectories starting $10^{-10}$ apart diverge exponentially.
• Strange attractor: trajectories are bounded but never repeat — they lie on a fractal set with dimension $\approx 2.06$.
• Butterfly shape: the attractor has two lobes around the two unstable fixed points at $(\pm\sqrt{\beta(\rho-1)}, \pm\sqrt{\beta(\rho-1)}, \rho-1)$.

Your Task

Implement:

• lorenz_deriv(state, sigma=10, rho=28, beta=8/3) — returns $(dx, dy, dz)$ as a tuple
• lorenz_rk4_step(state, sigma=10, rho=28, beta=8/3, dt=0.01) — one RK4 step, returns new $(x, y, z)$
• lorenz_trajectory(x0, y0, z0, n_steps=1000, dt=0.01) — returns list of $(x,y,z)$ tuples (including initial state)