systems.builtin.deterministic.continuous.ControlledLorenz

systems.builtin.deterministic.continuous.ControlledLorenz(*args, **kwargs)

Lorenz system - famous chaotic dynamical system from atmospheric convection with a forcing term

Physical System:

A simplified model of atmospheric convection. The system models: - Fluid circulation in a heated layer between two plates - Rate of convective overturning (x) - Horizontal temperature variation (y) - Vertical temperature variation (z)

State Space:

State: x = [x, y, z] - x: Rate of convective motion [dimensionless] * x > 0: clockwise circulation * x < 0: counterclockwise circulation * Proportional to velocity of fluid flow

- y: Horizontal temperature variation [dimensionless]
  * y > 0: warmer on one side
  * y < 0: warmer on other side
  * Temperature difference driving convection

- z: Vertical temperature variation from linearity [dimensionless]
  * z > 0: more stratified (stable)
  * z < 0: less stratified (unstable)
  * Deviation from conductive temperature profile

Control: u = [u] - u: External forcing/perturbation [dimensionless] - Typically u = 0 for studying natural chaos - Can be used to control or suppress chaos

Output: y = [x, y] - Partial observation: measures x and y, not z - Models limited sensor availability - Creates observability challenges for state estimation

Dynamics:

The Lorenz equations with control:

ẋ = σ(y - x) + u
ẏ = x(ρ - z) - y
ż = xy - βz

First equation (convection rate): - σ(y - x): Proportional to temperature difference - σ (sigma): Prandtl number - ratio of viscosity to thermal diffusivity - Drives x toward y at rate σ - Control u added here for external forcing

Second equation (horizontal temperature): - x(ρ - z): Nonlinear coupling - convection affects temperature - ρ (rho): Rayleigh number - ratio of buoyancy to viscous forces - -y: Damping term (heat diffusion) - When z < ρ, convection x amplifies y

Third equation (vertical temperature): - xy: Nonlinear product - convection creates temperature gradients - -βz: Damping/relaxation toward linear profile - β (beta): Geometric factor (aspect ratio of convection cell)

Parameters:

sigma : float, default=10.0 Prandtl number [dimensionless]. Ratio of momentum diffusivity (viscosity) to thermal diffusivity. Standard Chaotic Lorenz: σ = 10 Higher σ → faster adjustment of x to y

rho : float, default=28.0 Rayleigh number [dimensionless]. Measures temperature difference driving convection relative to dissipative effects. Critical values: - ρ < 1: No convection (conduction only) - 1 < ρ < 24.74: Steady convection - ρ > 24.74: Chaotic behavior possible - ρ = 28: Classic chaotic Lorenz attractor Higher ρ → stronger driving force

beta : float, default=8/3 Geometric factor [dimensionless]. Related to aspect ratio of convection cell (width/height). Standard value 8/3 ≈ 2.667 gives the classic “butterfly” attractor shape. - Affects dissipation rate in z - Controls attractor shape and size

Equilibria:

Origin (unstable for ρ > 1): x_eq = [0, 0, 0] (no convection) u_eq = 0

Stable when ρ < 1 (conduction dominates). Unstable when ρ > 1 (convection develops).

Convective equilibria (for ρ > 1): C+ = [√(β(ρ-1)), √(β(ρ-1)), ρ-1] C- = [-√(β(ρ-1)), -√(β(ρ-1)), ρ-1]

These represent steady clockwise (C+) and counterclockwise (C-) convection cells. Both become unstable for ρ > 24.74, leading to chaos.

Behavior Regimes:

  1. ρ < 1 (No convection):
    • Origin is stable
    • All trajectories decay to zero
    • Heat transported by conduction only
  2. 1 < ρ < 13.926 (Steady convection):
    • Origin becomes unstable
    • C+ or C- are stable (bistable system)
    • Steady convection cells form
  3. 13.926 < ρ < 24.74 (Periodic/complex):
    • C+ and C- lose stability
    • Can have limit cycles or complex behavior
  4. ρ > 24.74 (Chaos):
    • Chaotic behavior emerges
    • Sensitive dependence on initial conditions
    • Strange attractor (Lorenz butterfly)
  5. ρ = 28 (Classic chaos):
    • Well-studied chaotic attractor
    • Fractal structure
    • Positive Lyapunov exponent

The Lorenz Attractor:

For standard parameters (σ=10, ρ=28, β=8/3): - Shape: Two wing-like lobes (butterfly shape) - Structure: Strange attractor (fractal dimension ≈ 2.06) - Behavior: Trajectories spiral around C+ or C-, occasionally switching between wings - Predictability: Initial condition error doubles ~every 2 time units - Volume contraction: Phase space volume shrinks → dissipative system

Physical Interpretation:

  • x: Velocity of convection roll
  • y: Temperature difference between ascending and descending fluid
  • z: Deviation from linear temperature profile
  • ρ: Driving force (heating from below)
  • σ: Fluid properties (viscosity vs. thermal conductivity)
  • β: Cell geometry

See Also:

DuffingOscillator : Another chaotic system (forced oscillator) VanDerPolOscillator : Limit cycle oscillator NonlinearChainSystem : Coupled oscillators with complex dynamics