Lorenz Attractor (Lorenz-v0)

The Lorenz environment implements the classic Lorenz system — a three-dimensional chaotic system which, at the standard parameters, has three unstable equilibria: the origin and the two non-trivial equilibria $(\pm\sqrt{\beta(\rho-1)},\, \pm\sqrt{\beta(\rho-1)},\, \rho - 1)$. The control task is to stabilise the state at the non-trivial equilibrium with positive $x_1, x_2$, $x^* = (\sqrt{\beta(\rho-1)},\, \sqrt{\beta(\rho-1)},\, \rho - 1)$.

Dynamics

$$\begin{aligned} \dot{x}_1 &= \sigma (x_2 - x_1) + u \\ \dot{x}_2 &= x_1 (\rho - x_3) - x_2 \\ \dot{x}_3 &= x_1 x_2 - \beta x_3 \end{aligned}$$

Standard Lorenz parameters: $\sigma = 10$, $\rho = 28$, $\beta = 8/3$.

The action $u$ enters additively through $\dot{x}_1$. Integration uses RK45 with $\Delta t = 0.01\,\text{s}$.

Spaces and limits

Property	Value
State dimension	3
Action dimension	1
$x_1$ range	$[-20, 20]$
$x_2$ range	$[-50, 50]$
$x_3$ range	$[0, 50]$
Action range	$[-75, 75]$
$\Delta t$	0.01 s
Max episode steps	2000 (20 s)

Reference point

$$x^* = \bigl(\sqrt{\beta(\rho-1)},\; \sqrt{\beta(\rho-1)},\; \rho - 1\bigr) \approx (8.485,\; 8.485,\; 27.0)$$

Cost function

$$c(x, u) = (x - x^*)^\top Q (x - x^*) + u^\top R u$$

with $Q = I_3$ and $R = 0.001 \cdot I_1$. The small action cost ($R = 0.001$) reflects the large control authority needed to overcome chaotic divergence.

LQR linearisation

The LQR baseline linearises around $x^*$:

$$A_{\text{lin}} = \begin{pmatrix} -\sigma & \sigma & 0 \\ \rho - z^* & -1 & 0 \\ y^* & x^* & -\beta \end{pmatrix}, \qquad B_{\text{lin}} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$

Usage

uv run -m koopmanrl.soft_koopman_value_iteration --env_id Lorenz-v0
uv run -m koopmanrl.soft_actor_koopman_critic \
    --config_file configurations/sakc_lorenz_hparams.json

Notes

• The Lorenz system is chaotic for the standard parameters; without control, trajectories diverge from $x^*$ exponentially.
• The large action range ($\pm 75$) is needed for reliable stabilisation from arbitrary initial conditions.
• Source: koopmanrl/environments/lorenz.py