systems.builtin.stochastic.continuous.StochasticLorenz

systems.builtin.stochastic.continuous.StochasticLorenz(*args, **kwargs)

Stochastic Lorenz system - deterministic chaos with random perturbations.

Combines the iconic chaotic Lorenz attractor with continuous Brownian noise, creating a fundamental model for understanding the interplay between deterministic chaos and stochastic fluctuations.

Stochastic Differential Equations

The stochastic Lorenz equations:

dx = σ·(y - x)·dt + σ_x·dW_x
dy = (x·(ρ - z) - y)·dt + σ_y·dW_y
dz = (x·y - β·z)·dt + σ_z·dW_z

where: - x: Convection intensity (horizontal circulation) [dimensionless] - y: Horizontal temperature variation [dimensionless] - z: Vertical temperature variation [dimensionless] - σ: Prandtl number (fluid property, typically 10) - ρ: Rayleigh number (driving force, typically 28 for chaos) - β: Geometric factor (aspect ratio, typically 8/3) - σ_x, σ_y, σ_z: Noise intensities [1/√s] - W_x, W_y, W_z: Independent Wiener processes

Deterministic Part: Classic Lorenz dynamics - creates strange attractor and chaos.

Stochastic Part: Additive noise representing: - Unresolved turbulent fluctuations - Measurement errors - Model uncertainty - External forcing variability

Physical Interpretation

Original Context: Rayleigh-Bénard Convection

Fluid layer heated from below: - x: Convection roll intensity - y: Temperature difference (horizontal) - z: Temperature difference (vertical)

Parameters: - σ = 10: Prandtl number (momentum/thermal diffusivity) - ρ = 28: Rayleigh number (buoyancy/viscosity) - β = 8/3: Geometry (cell aspect ratio)

Atmospheric Interpretation: - x: Zonal wind component - y: Meridional wind component - z: Temperature deviation

Noise Sources: - Turbulent eddies below grid scale - Latent heat release variations - Radiation fluctuations - Measurement uncertainty

Deterministic Chaos Primer

Strange Attractor:

For σ=10, ρ=28, β=8/3: - Butterfly-shaped attractor in 3D - Fractal dimension: D ≈ 2.06 - Trajectories never repeat - Sensitive to initial conditions

Lyapunov Exponents: - λ₁ ≈ +0.9: Positive (chaos!) - λ₂ ≈ 0: Zero (volume-preserving direction) - λ₃ ≈ -14.6: Negative (dissipation)

Predictability Time: - Doubling time: τ ≈ 1/λ₁ ≈ 1.1 Lorenz time units - After ~10 doubling times: Completely unpredictable

Equilibria: - Origin (0,0,0): Unstable for ρ > 1 - C± = (±√(β(ρ-1)), ±√(β(ρ-1)), ρ-1): Unstable for ρ > ρ_crit

All unstable → trajectories on attractor.

Effect of Noise on Chaos

Small Noise (σ_noise << 1): - Attractor recognizable (slightly blurred) - Lyapunov exponent nearly unchanged - Chaos dominates, noise is perturbation

Medium Noise (σ_noise ~ 1): - Attractor significantly blurred - Lyapunov exponent increases - Chaos and noise comparable

Large Noise (σ_noise >> 1): - Attractor destroyed (fills volume) - Noise dominates dynamics - Chaos irrelevant (randomness wins)

Crossover Scale: When σ_noise ~ √λ_max, chaos and noise equally important.

Fractal Dimension: Noise increases effective dimension: - Deterministic: D ≈ 2.06 (fractal) - With noise: D → 3 (fills space)

Predictability: Combined error growth: δ²(t) = δ_0²·exp(2λt) + (σ²/λ)·(exp(2λt) - 1)

Both exponential growth (chaos) and accumulation (noise).

Key Properties

1. Nonlinearity: Quadratic terms: x·y, x·z create complex dynamics.

2. Chaos (Deterministic Part): Positive Lyapunov exponent → sensitive dependence.

3. Strange Attractor: Fractal structure (deterministic) blurred by noise.

4. Non-Stationary Locally: On attractor, system never settles.

5. Stationary Globally: Probability distribution on attractor converges (Fokker-Planck).

6. Three-Dimensional: Minimum dimension for chaos (Poincaré-Bendixson theorem).

7. Additive Noise (This Implementation): Constant diffusion coefficients (simplest model).

Mathematical Properties

No Closed-Form Solution: Even deterministic Lorenz has no analytical solution. Must integrate numerically.

Fokker-Planck Equation:

Probability density p(x,y,z,t) satisfies: ∂p/∂t = -∇·(f·p) + (1/2)·Σᵢ ∂²((σᵢ²)·p)/∂xᵢ²

For stationary distribution: ∂p/∂t = 0

Stationary Distribution: Concentrated on noise-perturbed attractor: - No simple analytical form - Compute via long-time simulation (ergodic hypothesis)

Moments: No simple analytical expressions. Compute empirically from ensemble or long trajectory.

State Space

State: X = [x, y, z] ∈ ℝ³ - Unbounded (but attracted to finite region) - Chaotic wandering on strange attractor - Noise perturbs trajectories

Control: u (optional, not standard) - Can add control to stabilize UPOs - Chaos control applications

Noise: W = [W_x, W_y, W_z] ∈ ℝ³ - Three independent Wiener processes - Models unresolved fluctuations

Parameters

Name	Type	Description	Default
sigma	float	Prandtl number (deterministic parameter) - Controls x-y coupling strength - Standard: 10	`10.0`
rho	float	Rayleigh number (deterministic parameter) - Controls bifurcations - ρ < 1: Stable origin - 1 < ρ < 24.74: Stable C± - ρ > 24.74: Chaos - Standard: 28 (chaotic regime)	`28.0`
beta	float	Geometric factor (deterministic parameter) - Standard: 8/3 ≈ 2.667	`8/3`
sigma_x	float	Noise intensity for x [1/√s]	`0.1`
sigma_y	float	Noise intensity for y [1/√s]	`0.1`
sigma_z	float	Noise intensity for z [1/√s]	`0.1`

Stochastic Properties

System Type: NONLINEAR (quadratic)
Noise Type: ADDITIVE (constant)
SDE Type: Itô
Noise Dimension: nw = 3
Chaotic: Yes (deterministic part)
Stationary: Yes (global, on attractor)
Ergodic: Yes (time = ensemble averages)

Applications

1. Meteorology: - Weather prediction limits - Ensemble forecasting - Data assimilation (EnKF, 4D-Var)

2. Climate Science: - Chaos vs variability - Tipping points under noise - Extreme events

3. Chaos Theory: - Noise effects on strange attractors - Modified Lyapunov exponents - Stochastic bifurcations

4. Signal Processing: - Chaotic time series analysis - Noise reduction - Chaos vs noise discrimination

5. Nonlinear Filtering: - Benchmark for EnKF, particle filters - Chaotic dynamics challenge

6. Control: - Chaos control under noise - Targeting unstable periodic orbits - Synchronization

Numerical Integration

Recommended: - Euler-Maruyama: dt = 0.001-0.01 - Smaller than deterministic (chaos + noise) - Check attractor structure

Validation: - Small noise: Should resemble Lorenz attractor - Statistics: Mean, variance converge - Lyapunov: Positive λ₁ (chaos indicator)

Monte Carlo Analysis

Ensemble Methods:

Run N = 100-1,000 trajectories: - Mean trajectory (climbs to attractor) - Spread on attractor - Long-time statistics

Attractor Visualization: - 3D scatter plot of all trajectories - Should show butterfly shape (if noise small)

Comparison with Deterministic

Deterministic Lorenz: - Pure chaos (no external randomness) - Clean strange attractor - Exponential divergence only

Stochastic Lorenz: - Chaos + noise combined - Blurred attractor - Exponential + diffusive error growth

When Stochastic Needed: - Real data has noise (always) - Predictability limits more realistic - Data assimilation applications

Limitations

Additive noise only
Constant noise (not state-dependent)
Independent noise (no correlation)
No control (in standard form)

Extensions

Multiplicative noise: σᵢ(X)
Colored noise: OU driving processes
Coupled Lorenz systems
Controlled Lorenz
Stochastic Rössler system

Methods

Name	Description
compute_attractor_size	Estimate typical attractor size (order of magnitude).
define_system	Define stochastic Lorenz system dynamics.
get_noise_intensities	Get noise intensity parameters.
setup_equilibria	Set up equilibrium points (deterministic part).

compute_attractor_size

systems.builtin.stochastic.continuous.StochasticLorenz.compute_attractor_size()

Estimate typical attractor size (order of magnitude).

Returns

Name	Type	Description
	float	Approximate attractor diameter

Notes

Rough estimate based on C± equilibria distance from origin.

Examples

>>> lorenz = StochasticLorenz(rho=28.0, beta=8/3)
>>> size = lorenz.compute_attractor_size()
>>> print(f"Attractor size: ~{size:.1f}")

define_system

systems.builtin.stochastic.continuous.StochasticLorenz.define_system(
    sigma=10.0,
    rho=28.0,
    beta=8.0 / 3.0,
    sigma_x=0.1,
    sigma_y=0.1,
    sigma_z=0.1,
)

Define stochastic Lorenz system dynamics.

Parameters

Name	Type	Description	Default
sigma	float	Prandtl number (deterministic parameter) - Controls x-y coupling - Standard: 10 - Ratio of momentum/thermal diffusivity	`10.0`
rho	float	Rayleigh number (deterministic parameter) - Controls convection strength - ρ < 1: Stable origin (no convection) - 1 < ρ < 24.74: Stable C± (steady convection) - ρ > 24.74: Chaos (turbulent convection) - Standard: 28 (chaotic regime)	`28.0`
beta	float	Geometric factor (deterministic parameter) - Standard: 8/3 ≈ 2.667 - Related to cell aspect ratio	`8/3`
sigma_x	float	Noise intensity for x [1/√s] - Unresolved turbulence - Typical: 0.01-1.0	`0.1`
sigma_y	float	Noise intensity for y [1/√s]	`0.1`
sigma_z	float	Noise intensity for z [1/√s]	`0.1`

Notes

Parameter Regimes:

ρ (Rayleigh Number): - ρ < 1: No convection (stable origin) - 1 < ρ < 13.93: Steady convection (stable C±) - 13.93 < ρ < 24.06: Transient chaos - 24.74 < ρ < 30: Strange attractor (standard) - ρ > 30: Different attractor structures

Classic Chaotic: σ = 10, ρ = 28, β = 8/3

Noise Intensity Guidelines:

Relative to chaos strength: - σ_noise < 0.1: Noise negligible (chaos dominates) - σ_noise ~ 0.1-1.0: Noise perturbs chaos (typical) - σ_noise > 1.0: Noise dominates (chaos obscured)

Physical Interpretation:

For atmospheric application: - σ_noise represents unresolved processes - Should be calibrated from data (residuals) - Typical: 0.1-0.5 for simplified models

Noise Correlation:

Currently: Independent noise on each variable.

In reality: May be correlated (common weather systems). Extension: Use full 3×3 covariance matrix.

Chaos Indicators:

Check if deterministic part is chaotic: - Compute Lyapunov exponents from noise-free simulation - λ₁ > 0 indicates chaos - For ρ = 28: λ₁ ≈ 0.9 (strongly chaotic)

Noise Impact on Lyapunov:

Noise increases apparent Lyapunov exponent: λ_apparent ≈ λ_det + σ²/(2·scale²)

where scale is attractor size.

get_noise_intensities

systems.builtin.stochastic.continuous.StochasticLorenz.get_noise_intensities()

Get noise intensity parameters.

Returns

Name	Type	Description
	dict	{‘sigma_x’: …, ‘sigma_y’: …, ‘sigma_z’: …}

setup_equilibria

systems.builtin.stochastic.continuous.StochasticLorenz.setup_equilibria()

Set up equilibrium points (deterministic part).

Note: All equilibria are unstable in chaotic regime (ρ=28). Trajectories never settle but wander on attractor.

# systems.builtin.stochastic.continuous.StochasticLorenz { #cdesym.systems.builtin.stochastic.continuous.StochasticLorenz } ```python systems.builtin.stochastic.continuous.StochasticLorenz(*args, **kwargs) ``` Stochastic Lorenz system - deterministic chaos with random perturbations. Combines the iconic chaotic Lorenz attractor with continuous Brownian noise, creating a fundamental model for understanding the interplay between deterministic chaos and stochastic fluctuations. ## Stochastic Differential Equations {.doc-section .doc-section-stochastic-differential-equations} The stochastic Lorenz equations: dx = σ·(y - x)·dt + σ_x·dW_x dy = (x·(ρ - z) - y)·dt + σ_y·dW_y dz = (x·y - β·z)·dt + σ_z·dW_z where: - x: Convection intensity (horizontal circulation) [dimensionless] - y: Horizontal temperature variation [dimensionless] - z: Vertical temperature variation [dimensionless] - σ: Prandtl number (fluid property, typically 10) - ρ: Rayleigh number (driving force, typically 28 for chaos) - β: Geometric factor (aspect ratio, typically 8/3) - σ_x, σ_y, σ_z: Noise intensities [1/√s] - W_x, W_y, W_z: Independent Wiener processes **Deterministic Part:** Classic Lorenz dynamics - creates strange attractor and chaos. **Stochastic Part:** Additive noise representing: - Unresolved turbulent fluctuations - Measurement errors - Model uncertainty - External forcing variability ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Original Context: Rayleigh-Bénard Convection** Fluid layer heated from below: - x: Convection roll intensity - y: Temperature difference (horizontal) - z: Temperature difference (vertical) **Parameters:** - σ = 10: Prandtl number (momentum/thermal diffusivity) - ρ = 28: Rayleigh number (buoyancy/viscosity) - β = 8/3: Geometry (cell aspect ratio) **Atmospheric Interpretation:** - x: Zonal wind component - y: Meridional wind component - z: Temperature deviation **Noise Sources:** - Turbulent eddies below grid scale - Latent heat release variations - Radiation fluctuations - Measurement uncertainty ## Deterministic Chaos Primer {.doc-section .doc-section-deterministic-chaos-primer} **Strange Attractor:** For σ=10, ρ=28, β=8/3: - Butterfly-shaped attractor in 3D - Fractal dimension: D ≈ 2.06 - Trajectories never repeat - Sensitive to initial conditions **Lyapunov Exponents:** - λ₁ ≈ +0.9: Positive (chaos!) - λ₂ ≈ 0: Zero (volume-preserving direction) - λ₃ ≈ -14.6: Negative (dissipation) **Predictability Time:** - Doubling time: τ ≈ 1/λ₁ ≈ 1.1 Lorenz time units - After ~10 doubling times: Completely unpredictable **Equilibria:** - Origin (0,0,0): Unstable for ρ > 1 - C± = (±√(β(ρ-1)), ±√(β(ρ-1)), ρ-1): Unstable for ρ > ρ_crit All unstable → trajectories on attractor. ## Effect of Noise on Chaos {.doc-section .doc-section-effect-of-noise-on-chaos} **Small Noise (σ_noise << 1):** - Attractor recognizable (slightly blurred) - Lyapunov exponent nearly unchanged - Chaos dominates, noise is perturbation **Medium Noise (σ_noise ~ 1):** - Attractor significantly blurred - Lyapunov exponent increases - Chaos and noise comparable **Large Noise (σ_noise >> 1):** - Attractor destroyed (fills volume) - Noise dominates dynamics - Chaos irrelevant (randomness wins) **Crossover Scale:** When σ_noise ~ √λ_max, chaos and noise equally important. **Fractal Dimension:** Noise increases effective dimension: - Deterministic: D ≈ 2.06 (fractal) - With noise: D → 3 (fills space) **Predictability:** Combined error growth: δ²(t) = δ_0²·exp(2λt) + (σ²/λ)·(exp(2λt) - 1) Both exponential growth (chaos) and accumulation (noise). ## Key Properties {.doc-section .doc-section-key-properties} **1. Nonlinearity:** Quadratic terms: x·y, x·z create complex dynamics. **2. Chaos (Deterministic Part):** Positive Lyapunov exponent → sensitive dependence. **3. Strange Attractor:** Fractal structure (deterministic) blurred by noise. **4. Non-Stationary Locally:** On attractor, system never settles. **5. Stationary Globally:** Probability distribution on attractor converges (Fokker-Planck). **6. Three-Dimensional:** Minimum dimension for chaos (Poincaré-Bendixson theorem). **7. Additive Noise (This Implementation):** Constant diffusion coefficients (simplest model). ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **No Closed-Form Solution:** Even deterministic Lorenz has no analytical solution. Must integrate numerically. **Fokker-Planck Equation:** Probability density p(x,y,z,t) satisfies: ∂p/∂t = -∇·(f·p) + (1/2)·Σᵢ ∂²((σᵢ²)·p)/∂xᵢ² For stationary distribution: ∂p/∂t = 0 **Stationary Distribution:** Concentrated on noise-perturbed attractor: - No simple analytical form - Compute via long-time simulation (ergodic hypothesis) **Moments:** No simple analytical expressions. Compute empirically from ensemble or long trajectory. ## State Space {.doc-section .doc-section-state-space} State: X = [x, y, z] ∈ ℝ³ - Unbounded (but attracted to finite region) - Chaotic wandering on strange attractor - Noise perturbs trajectories Control: u (optional, not standard) - Can add control to stabilize UPOs - Chaos control applications Noise: W = [W_x, W_y, W_z] ∈ ℝ³ - Three independent Wiener processes - Models unresolved fluctuations ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------|--------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | sigma | float | Prandtl number (deterministic parameter) - Controls x-y coupling strength - Standard: 10 | `10.0` | | rho | float | Rayleigh number (deterministic parameter) - Controls bifurcations - ρ < 1: Stable origin - 1 < ρ < 24.74: Stable C± - ρ > 24.74: Chaos - Standard: 28 (chaotic regime) | `28.0` | | beta | float | Geometric factor (deterministic parameter) - Standard: 8/3 ≈ 2.667 | `8/3` | | sigma_x | float | Noise intensity for x [1/√s] | `0.1` | | sigma_y | float | Noise intensity for y [1/√s] | `0.1` | | sigma_z | float | Noise intensity for z [1/√s] | `0.1` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - System Type: NONLINEAR (quadratic) - Noise Type: ADDITIVE (constant) - SDE Type: Itô - Noise Dimension: nw = 3 - Chaotic: Yes (deterministic part) - Stationary: Yes (global, on attractor) - Ergodic: Yes (time = ensemble averages) ## Applications {.doc-section .doc-section-applications} **1. Meteorology:** - Weather prediction limits - Ensemble forecasting - Data assimilation (EnKF, 4D-Var) **2. Climate Science:** - Chaos vs variability - Tipping points under noise - Extreme events **3. Chaos Theory:** - Noise effects on strange attractors - Modified Lyapunov exponents - Stochastic bifurcations **4. Signal Processing:** - Chaotic time series analysis - Noise reduction - Chaos vs noise discrimination **5. Nonlinear Filtering:** - Benchmark for EnKF, particle filters - Chaotic dynamics challenge **6. Control:** - Chaos control under noise - Targeting unstable periodic orbits - Synchronization ## Numerical Integration {.doc-section .doc-section-numerical-integration} **Recommended:** - Euler-Maruyama: dt = 0.001-0.01 - Smaller than deterministic (chaos + noise) - Check attractor structure **Validation:** - Small noise: Should resemble Lorenz attractor - Statistics: Mean, variance converge - Lyapunov: Positive λ₁ (chaos indicator) ## Monte Carlo Analysis {.doc-section .doc-section-monte-carlo-analysis} **Ensemble Methods:** Run N = 100-1,000 trajectories: - Mean trajectory (climbs to attractor) - Spread on attractor - Long-time statistics **Attractor Visualization:** - 3D scatter plot of all trajectories - Should show butterfly shape (if noise small) ## Comparison with Deterministic {.doc-section .doc-section-comparison-with-deterministic} **Deterministic Lorenz:** - Pure chaos (no external randomness) - Clean strange attractor - Exponential divergence only **Stochastic Lorenz:** - Chaos + noise combined - Blurred attractor - Exponential + diffusive error growth **When Stochastic Needed:** - Real data has noise (always) - Predictability limits more realistic - Data assimilation applications ## Limitations {.doc-section .doc-section-limitations} - Additive noise only - Constant noise (not state-dependent) - Independent noise (no correlation) - No control (in standard form) ## Extensions {.doc-section .doc-section-extensions} - Multiplicative noise: σᵢ(X) - Colored noise: OU driving processes - Coupled Lorenz systems - Controlled Lorenz - Stochastic Rössler system ## See Also {.doc-section .doc-section-see-also} Lorenz : Deterministic version ## Methods | Name | Description | | --- | --- | | [compute_attractor_size](#cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.compute_attractor_size) | Estimate typical attractor size (order of magnitude). | | [define_system](#cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.define_system) | Define stochastic Lorenz system dynamics. | | [get_noise_intensities](#cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.get_noise_intensities) | Get noise intensity parameters. | | [setup_equilibria](#cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.setup_equilibria) | Set up equilibrium points (deterministic part). | ### compute_attractor_size { #cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.compute_attractor_size } ```python systems.builtin.stochastic.continuous.StochasticLorenz.compute_attractor_size() ``` Estimate typical attractor size (order of magnitude). #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|--------------------------------| | | float | Approximate attractor diameter | #### Notes {.doc-section .doc-section-notes} Rough estimate based on C± equilibria distance from origin. #### Examples {.doc-section .doc-section-examples} ```python >>> lorenz = StochasticLorenz(rho=28.0, beta=8/3) >>> size = lorenz.compute_attractor_size() >>> print(f"Attractor size: ~{size:.1f}") ``` ### define_system { #cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.define_system } ```python systems.builtin.stochastic.continuous.StochasticLorenz.define_system( sigma=10.0, rho=28.0, beta=8.0 / 3.0, sigma_x=0.1, sigma_y=0.1, sigma_z=0.1, ) ``` Define stochastic Lorenz system dynamics. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------|--------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | sigma | float | Prandtl number (deterministic parameter) - Controls x-y coupling - Standard: 10 - Ratio of momentum/thermal diffusivity | `10.0` | | rho | float | Rayleigh number (deterministic parameter) - Controls convection strength - ρ < 1: Stable origin (no convection) - 1 < ρ < 24.74: Stable C± (steady convection) - ρ > 24.74: Chaos (turbulent convection) - Standard: 28 (chaotic regime) | `28.0` | | beta | float | Geometric factor (deterministic parameter) - Standard: 8/3 ≈ 2.667 - Related to cell aspect ratio | `8/3` | | sigma_x | float | Noise intensity for x [1/√s] - Unresolved turbulence - Typical: 0.01-1.0 | `0.1` | | sigma_y | float | Noise intensity for y [1/√s] | `0.1` | | sigma_z | float | Noise intensity for z [1/√s] | `0.1` | #### Notes {.doc-section .doc-section-notes} **Parameter Regimes:** **ρ (Rayleigh Number):** - ρ < 1: No convection (stable origin) - 1 < ρ < 13.93: Steady convection (stable C±) - 13.93 < ρ < 24.06: Transient chaos - 24.74 < ρ < 30: Strange attractor (standard) - ρ > 30: Different attractor structures **Classic Chaotic:** σ = 10, ρ = 28, β = 8/3 **Noise Intensity Guidelines:** Relative to chaos strength: - σ_noise < 0.1: Noise negligible (chaos dominates) - σ_noise ~ 0.1-1.0: Noise perturbs chaos (typical) - σ_noise > 1.0: Noise dominates (chaos obscured) **Physical Interpretation:** For atmospheric application: - σ_noise represents unresolved processes - Should be calibrated from data (residuals) - Typical: 0.1-0.5 for simplified models **Noise Correlation:** Currently: Independent noise on each variable. In reality: May be correlated (common weather systems). Extension: Use full 3×3 covariance matrix. **Chaos Indicators:** Check if deterministic part is chaotic: - Compute Lyapunov exponents from noise-free simulation - λ₁ > 0 indicates chaos - For ρ = 28: λ₁ ≈ 0.9 (strongly chaotic) **Noise Impact on Lyapunov:** Noise increases apparent Lyapunov exponent: λ_apparent ≈ λ_det + σ²/(2·scale²) where scale is attractor size. ### get_noise_intensities { #cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.get_noise_intensities } ```python systems.builtin.stochastic.continuous.StochasticLorenz.get_noise_intensities() ``` Get noise intensity parameters. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|--------------------------------------------------| | | dict | {'sigma_x': ..., 'sigma_y': ..., 'sigma_z': ...} | ### setup_equilibria { #cdesym.systems.builtin.stochastic.continuous.StochasticLorenz.setup_equilibria } ```python systems.builtin.stochastic.continuous.StochasticLorenz.setup_equilibria() ``` Set up equilibrium points (deterministic part). Note: All equilibria are unstable in chaotic regime (ρ=28). Trajectories never settle but wander on attractor.