systems.builtin.stochastic.continuous.ContinuousStochasticPendulum

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

Stochastic pendulum with random forcing - canonical nonlinear stochastic system.

Combines the nonlinear dynamics of a simple pendulum with continuous random forcing, creating a fundamental model for studying noise-induced phenomena, Kramers’ escape theory, stochastic resonance, and nonlinear stochastic control.

Stochastic Differential Equations

State-space form: dθ = ω·dt dω = (-(g/L)·sin(θ) - b·ω + u)·dt + σ·dW

Normalized form (g/L = 1): dθ = ω·dt dω = (-sin(θ) - b·ω + u)·dt + σ·dW

where: - θ: Angle from downward vertical [rad] - ω: Angular velocity [rad/s] - g/L: Gravitational constant (normalized to 1) - b: Damping coefficient [1/s] - u: Applied torque [rad/s²] - σ: Noise intensity [rad/(s²·√s)] - W(t): Standard Wiener process

Physical Interpretation

Equation of Motion:

Newton’s rotational law: I·θ̈ = τ_total

For unit moment of inertia (I=1): θ̈ = τ_gravity + τ_damping + τ_control + τ_noise θ̈ = -(g/L)·sin(θ) - b·θ̇ + u + σ·ξ(t)

where ξ(t) is white noise (formal derivative of W).

Torque Components: 1. Gravity: -(g/L)·sin(θ) - Restoring torque toward θ=0 2. Damping: -b·ω - Dissipative (energy removal) 3. Control: u - Applied torque 4. Noise: σ·dW/dt - Random disturbances

Energy: E = (1/2)·ω² + (g/L)·(1 - cos(θ))

Kinetic + potential (normalized).

Noise Sources: - Mechanical: Air currents, vibrations - Thermal: Brownian motion at pivot - Model uncertainty: Unmodeled dynamics

Key Features

Nonlinearity: sin(θ) creates fundamentally different behavior from linear oscillator.

Bistability: Two potential wells (θ = 0, ±2π, …). Noise causes transitions.

Energy Barrier: ΔE = 2·g/L separates downward from inverted.

Additive Noise: Constant σ (state-independent). Enters velocity equation (physical forces).

Damping: b > 0 required for stationary distribution. Without damping: Energy fluctuates, no equilibrium.

Mathematical Properties

Small Angle Approximation (|θ| << 1):

sin(θ) ≈ θ gives linearized SDE: dθ = ω·dt dω = (-(g/L)·θ - b·ω + u)·dt + σ·dW

This is linear (harmonic oscillator with noise).

Stationary Distribution (b > 0, u = 0):

For damped pendulum: p_∞(θ, ω) ∝ exp(-E(θ,ω)/(σ²/b))

where E = (1/2)·ω² - (g/L)·cos(θ).

Effective Temperature: T_eff = σ²/b

Ratio of noise injection to dissipation.

Kramers’ Escape Rate:

From θ = 0 over barrier to θ = π: k_escape ≈ (ω_well²/2πb)·exp(-ΔE·b/σ²)

Exponentially small for σ << √(ΔE·b).

Physical Interpretation

Damping b: - Units: [1/s] - Controls energy dissipation - Time constant: τ = 1/b - Quality factor: Q = (g/L)^(1/2)/b

Examples: - b = 0.1: Underdamped (Q = 10) - b = 1.0: Moderate damping - b = 10.0: Overdamped (creeping motion)

Noise Intensity σ: - Units: [rad/(s²·√s)] - Controls random torque magnitude - Effective temperature: σ²/b

Examples: - σ = 0.1: Small noise (rare escapes) - σ = 0.5: Moderate (occasional escapes) - σ = 1.0: Large (frequent escapes)

Ratio σ/√b: Dimensionless noise level: - Small: Deep potential wells, rare transitions - Large: Shallow effective wells, frequent transitions

State Space

State: X = [θ, ω] - θ ∈ ℝ or S¹: Angle (unwrapped or periodic) - ω ∈ ℝ: Angular velocity

Control: u ∈ ℝ - Applied torque

Noise: w ∈ ℝ - Single Wiener process - Enters velocity equation

Parameters

Name	Type	Description	Default
g	float	Gravitational acceleration [m/s²]	`9.81`
L	float	Pendulum length [m]	`1.0`
b	float	Damping coefficient [1/s] - b > 0 for stationary distribution - Typical: 0.1-10.0	`0.5`
sigma	float	Noise intensity [rad/(s²·√s)] - Controls random torque - Typical: 0.1-2.0	`0.5`
m	float	Mass [kg] (optional, for dimensional analysis)	`1.0`

Stochastic Properties

System Type: NONLINEAR
Noise Type: ADDITIVE
SDE Type: Itô
Noise Dimension: nw = 1
Stationary: Yes (if b > 0)
Ergodic: Yes (if b > 0)
Bistable: Yes (multiple wells)

Applications

1. Kramers’ Escape Theory: - Noise-activated barrier crossing - Chemical reaction rates - Exponential escape time

2. Stochastic Resonance: - Weak periodic signal enhanced by noise - Optimal noise level exists - Biology, climate, electronics

3. Coherence Resonance: - Noise creates coherent oscillations - No periodic forcing needed - Maximum regularity at optimal σ

4. Nonlinear Control: - Swing-up with noise - Stabilization at inverted - Robust control design

5. Synchronization: - Coupled oscillators - Common noise induces correlation - Phase locking

Numerical Integration

Recommended: - Euler-Maruyama: dt = 0.01-0.1 s - Check angle wrapping (if needed) - Monitor energy (should fluctuate)

Convergence: - Weak: O(dt) for moments - Strong: O(√dt) for paths

Kramers Escape Analysis

Escape Time: Mean time to cross barrier: τ ~ exp(ΔE·b/σ²)

Exponentially sensitive to: - Barrier height ΔE - Damping b (appears in exponent!) - Noise σ² (inverse in exponent)

Comparison with Linear Oscillator

Linear (Harmonic): - sin(θ) ≈ θ - Gaussian stationary distribution - No bistability

Nonlinear (Pendulum): - Full sin(θ) - Non-Gaussian stationary distribution - Bistable (multiple wells)

Limitations

1D angle (no spatial motion)
Additive noise only
Constant parameters
No joint flexibility
Rigid body assumption

Extensions

Double pendulum (chaos + noise)
Elastic pendulum (variable length)
Spherical pendulum (2D angle)
Coupled pendulums (arrays)
Parametric noise (variable g or L)

Methods

Name	Description
compute_energy	Compute mechanical energy.
define_system	Define stochastic pendulum dynamics.
estimate_kramers_escape_time	Estimate mean escape time using Kramers formula.
get_effective_temperature	Get effective temperature T_eff = σ²/b.
get_natural_frequency	Get natural frequency ω_0 = √(g/L).
setup_equilibria	Set up equilibrium points (deterministic part).

compute_energy

systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.compute_energy(
    theta,
    omega,
)

Compute mechanical energy.

E = (1/2)·ω² + (g/L)·(1 - cos(θ))

Parameters

Name	Type	Description	Default
theta	float	Angle [rad]	required
omega	float	Angular velocity [rad/s]	required

Returns

Name	Type	Description
	float	Energy [J/kg] (or normalized)

Notes

Energy fluctuates randomly due to noise. With damping: Mean energy dissipates to T_eff = σ²/b.

Examples

>>> pend = ContinuousStochasticPendulum()
>>> E = pend.compute_energy(theta=0.5, omega=1.0)
>>> print(f"Energy: {E:.3f}")

define_system

systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.define_system(
    g=9.81,
    L=1.0,
    b=0.5,
    sigma=0.5,
    m=1.0,
)

Define stochastic pendulum dynamics.

Parameters

Name	Type	Description	Default
g	float	Gravitational acceleration [m/s²]	`9.81`
L	float	Pendulum length [m]	`1.0`
b	float	Damping coefficient [1/s] - b > 0 required for stationary distribution - Typical: 0.1-10.0 - Quality factor: Q = √(g/L)/b	`0.5`
sigma	float	Noise intensity [rad/(s²·√s)] - Controls random torque magnitude - Typical: 0.1-2.0 - Effective temperature: T_eff = σ²/b	`0.5`
m	float	Mass [kg] (optional, for dimensional analysis)	`1.0`

Notes

Natural Frequency: ω_0 = √(g/L)

For g=9.81, L=1: ω_0 ≈ 3.13 rad/s, Period ≈ 2 s

Damping Regimes:

Damping ratio: ζ = b/(2·ω_0) - ζ < 1: Underdamped (oscillatory decay) - ζ = 1: Critically damped (fastest non-oscillatory) - ζ > 1: Overdamped (slow exponential decay)

Effective Temperature: T_eff = σ²/b

Analogy with thermodynamics: - Noise injects energy: ~ σ² - Damping removes energy: ~ b - Equilibrium: Balance at T_eff

Energy Barrier: From stable (θ=0) to unstable (θ=π): ΔE = 2·g/L

For g=9.81, L=1: ΔE = 19.62 J/kg

Kramers Time: Mean escape time: τ_escape ~ (b/ω_0²)·exp(ΔE·b/σ²)

Example: b=0.5, σ=0.5, ΔE≈20 → τ ~ 0.05·exp(20) ~ 2.4×10⁸ seconds!

Noise Level Guidelines:

Low noise (σ < 0.3): - Rare escapes (τ > hours) - Nearly deterministic - Small fluctuations around equilibrium

Moderate noise (σ = 0.3-1.0): - Occasional escapes (τ ~ minutes to hours) - Visible stochastic effects - Coherence resonance regime

High noise (σ > 1.0): - Frequent escapes (τ ~ seconds) - Dominated by randomness - Large fluctuations

Stochastic Resonance Regime: For periodic forcing u = A·cos(ω·t): - Optimal σ ≈ √(ΔE·b) ≈ √(2·g/L·b) - Maximizes response to weak signal

estimate_kramers_escape_time

systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.estimate_kramers_escape_time(
)

Estimate mean escape time using Kramers formula.

τ ~ (2πb/ω_0²)·exp(ΔE·b/σ²)

Returns

Name	Type	Description
	float	Estimated mean escape time [s]

Notes

This is approximate, valid for: - Small noise: σ² << ΔE·b - Moderate damping: b ~ ω_0

For accurate times, use Monte Carlo simulation.

Examples

>>> pend = ContinuousStochasticPendulum(b=0.5, sigma=0.5)
>>> tau = pend.estimate_kramers_escape_time()
>>> print(f"Mean escape time: {tau:.2e} s")

get_effective_temperature

systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_effective_temperature(
)

Get effective temperature T_eff = σ²/b.

Ratio of noise injection to dissipation.

Returns

Name	Type	Description
	float	Effective temperature

Notes

Analogy with thermodynamics: - Higher T_eff: More energetic fluctuations - Lower T_eff: Smaller fluctuations - Controls stationary distribution width

Examples

>>> pend = ContinuousStochasticPendulum(b=0.5, sigma=0.5)
>>> T_eff = pend.get_effective_temperature()
>>> print(f"T_eff: {T_eff:.3f}")

get_natural_frequency

systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_natural_frequency(
)

Get natural frequency ω_0 = √(g/L).

Returns

Name	Type	Description
	float	Natural frequency [rad/s]

Examples

>>> pend = ContinuousStochasticPendulum(g=9.81, L=1.0)
>>> omega_0 = pend.get_natural_frequency()
>>> period = 2 * np.pi / omega_0
>>> print(f"Period: {period:.2f} s")

setup_equilibria

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

Set up equilibrium points (deterministic part).

Pendulum has periodic equilibria at θ = n·π.

# systems.builtin.stochastic.continuous.ContinuousStochasticPendulum { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum( *args, **kwargs, ) ``` Stochastic pendulum with random forcing - canonical nonlinear stochastic system. Combines the nonlinear dynamics of a simple pendulum with continuous random forcing, creating a fundamental model for studying noise-induced phenomena, Kramers' escape theory, stochastic resonance, and nonlinear stochastic control. ## Stochastic Differential Equations {.doc-section .doc-section-stochastic-differential-equations} State-space form: dθ = ω·dt dω = (-(g/L)·sin(θ) - b·ω + u)·dt + σ·dW Normalized form (g/L = 1): dθ = ω·dt dω = (-sin(θ) - b·ω + u)·dt + σ·dW where: - θ: Angle from downward vertical [rad] - ω: Angular velocity [rad/s] - g/L: Gravitational constant (normalized to 1) - b: Damping coefficient [1/s] - u: Applied torque [rad/s²] - σ: Noise intensity [rad/(s²·√s)] - W(t): Standard Wiener process ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Equation of Motion:** Newton's rotational law: I·θ̈ = τ_total For unit moment of inertia (I=1): θ̈ = τ_gravity + τ_damping + τ_control + τ_noise θ̈ = -(g/L)·sin(θ) - b·θ̇ + u + σ·ξ(t) where ξ(t) is white noise (formal derivative of W). **Torque Components:** 1. Gravity: -(g/L)·sin(θ) - Restoring torque toward θ=0 2. Damping: -b·ω - Dissipative (energy removal) 3. Control: u - Applied torque 4. Noise: σ·dW/dt - Random disturbances **Energy:** E = (1/2)·ω² + (g/L)·(1 - cos(θ)) Kinetic + potential (normalized). **Noise Sources:** - Mechanical: Air currents, vibrations - Thermal: Brownian motion at pivot - Model uncertainty: Unmodeled dynamics ## Key Features {.doc-section .doc-section-key-features} **Nonlinearity:** sin(θ) creates fundamentally different behavior from linear oscillator. **Bistability:** Two potential wells (θ = 0, ±2π, ...). Noise causes transitions. **Energy Barrier:** ΔE = 2·g/L separates downward from inverted. **Additive Noise:** Constant σ (state-independent). Enters velocity equation (physical forces). **Damping:** b > 0 required for stationary distribution. Without damping: Energy fluctuates, no equilibrium. ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Small Angle Approximation (|θ| << 1):** sin(θ) ≈ θ gives linearized SDE: dθ = ω·dt dω = (-(g/L)·θ - b·ω + u)·dt + σ·dW This is linear (harmonic oscillator with noise). **Stationary Distribution (b > 0, u = 0):** For damped pendulum: p_∞(θ, ω) ∝ exp(-E(θ,ω)/(σ²/b)) where E = (1/2)·ω² - (g/L)·cos(θ). **Effective Temperature:** T_eff = σ²/b Ratio of noise injection to dissipation. **Kramers' Escape Rate:** From θ = 0 over barrier to θ = π: k_escape ≈ (ω_well²/2πb)·exp(-ΔE·b/σ²) Exponentially small for σ << √(ΔE·b). ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Damping b:** - Units: [1/s] - Controls energy dissipation - Time constant: τ = 1/b - Quality factor: Q = (g/L)^(1/2)/b **Examples:** - b = 0.1: Underdamped (Q = 10) - b = 1.0: Moderate damping - b = 10.0: Overdamped (creeping motion) **Noise Intensity σ:** - Units: [rad/(s²·√s)] - Controls random torque magnitude - Effective temperature: σ²/b **Examples:** - σ = 0.1: Small noise (rare escapes) - σ = 0.5: Moderate (occasional escapes) - σ = 1.0: Large (frequent escapes) **Ratio σ/√b:** Dimensionless noise level: - Small: Deep potential wells, rare transitions - Large: Shallow effective wells, frequent transitions ## State Space {.doc-section .doc-section-state-space} State: X = [θ, ω] - θ ∈ ℝ or S¹: Angle (unwrapped or periodic) - ω ∈ ℝ: Angular velocity Control: u ∈ ℝ - Applied torque Noise: w ∈ ℝ - Single Wiener process - Enters velocity equation ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-----------------------------------------------------------------------------------|-----------| | g | float | Gravitational acceleration [m/s²] | `9.81` | | L | float | Pendulum length [m] | `1.0` | | b | float | Damping coefficient [1/s] - b > 0 for stationary distribution - Typical: 0.1-10.0 | `0.5` | | sigma | float | Noise intensity [rad/(s²·√s)] - Controls random torque - Typical: 0.1-2.0 | `0.5` | | m | float | Mass [kg] (optional, for dimensional analysis) | `1.0` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - System Type: NONLINEAR - Noise Type: ADDITIVE - SDE Type: Itô - Noise Dimension: nw = 1 - Stationary: Yes (if b > 0) - Ergodic: Yes (if b > 0) - Bistable: Yes (multiple wells) ## Applications {.doc-section .doc-section-applications} **1. Kramers' Escape Theory:** - Noise-activated barrier crossing - Chemical reaction rates - Exponential escape time **2. Stochastic Resonance:** - Weak periodic signal enhanced by noise - Optimal noise level exists - Biology, climate, electronics **3. Coherence Resonance:** - Noise creates coherent oscillations - No periodic forcing needed - Maximum regularity at optimal σ **4. Nonlinear Control:** - Swing-up with noise - Stabilization at inverted - Robust control design **5. Synchronization:** - Coupled oscillators - Common noise induces correlation - Phase locking ## Numerical Integration {.doc-section .doc-section-numerical-integration} **Recommended:** - Euler-Maruyama: dt = 0.01-0.1 s - Check angle wrapping (if needed) - Monitor energy (should fluctuate) **Convergence:** - Weak: O(dt) for moments - Strong: O(√dt) for paths ## Kramers Escape Analysis {.doc-section .doc-section-kramers-escape-analysis} **Escape Time:** Mean time to cross barrier: τ ~ exp(ΔE·b/σ²) Exponentially sensitive to: - Barrier height ΔE - Damping b (appears in exponent!) - Noise σ² (inverse in exponent) ## Comparison with Linear Oscillator {.doc-section .doc-section-comparison-with-linear-oscillator} **Linear (Harmonic):** - sin(θ) ≈ θ - Gaussian stationary distribution - No bistability **Nonlinear (Pendulum):** - Full sin(θ) - Non-Gaussian stationary distribution - Bistable (multiple wells) ## Limitations {.doc-section .doc-section-limitations} - 1D angle (no spatial motion) - Additive noise only - Constant parameters - No joint flexibility - Rigid body assumption ## Extensions {.doc-section .doc-section-extensions} - Double pendulum (chaos + noise) - Elastic pendulum (variable length) - Spherical pendulum (2D angle) - Coupled pendulums (arrays) - Parametric noise (variable g or L) ## See Also {.doc-section .doc-section-see-also} Pendulum : Deterministic version StochasticDoubleIntegrator : Linear analog ## Methods | Name | Description | | --- | --- | | [compute_energy](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.compute_energy) | Compute mechanical energy. | | [define_system](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.define_system) | Define stochastic pendulum dynamics. | | [estimate_kramers_escape_time](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.estimate_kramers_escape_time) | Estimate mean escape time using Kramers formula. | | [get_effective_temperature](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_effective_temperature) | Get effective temperature T_eff = σ²/b. | | [get_natural_frequency](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_natural_frequency) | Get natural frequency ω_0 = √(g/L). | | [setup_equilibria](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.setup_equilibria) | Set up equilibrium points (deterministic part). | ### compute_energy { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.compute_energy } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.compute_energy( theta, omega, ) ``` Compute mechanical energy. E = (1/2)·ω² + (g/L)·(1 - cos(θ)) #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|--------------------------|------------| | theta | float | Angle [rad] | _required_ | | omega | float | Angular velocity [rad/s] | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-------------------------------| | | float | Energy [J/kg] (or normalized) | #### Notes {.doc-section .doc-section-notes} Energy fluctuates randomly due to noise. With damping: Mean energy dissipates to T_eff = σ²/b. #### Examples {.doc-section .doc-section-examples} ```python >>> pend = ContinuousStochasticPendulum() >>> E = pend.compute_energy(theta=0.5, omega=1.0) >>> print(f"Energy: {E:.3f}") ``` ### define_system { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.define_system } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.define_system( g=9.81, L=1.0, b=0.5, sigma=0.5, m=1.0, ) ``` Define stochastic pendulum dynamics. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|---------------------------------------------------------------------------------------------------------------------------|-----------| | g | float | Gravitational acceleration [m/s²] | `9.81` | | L | float | Pendulum length [m] | `1.0` | | b | float | Damping coefficient [1/s] - b > 0 required for stationary distribution - Typical: 0.1-10.0 - Quality factor: Q = √(g/L)/b | `0.5` | | sigma | float | Noise intensity [rad/(s²·√s)] - Controls random torque magnitude - Typical: 0.1-2.0 - Effective temperature: T_eff = σ²/b | `0.5` | | m | float | Mass [kg] (optional, for dimensional analysis) | `1.0` | #### Notes {.doc-section .doc-section-notes} **Natural Frequency:** ω_0 = √(g/L) For g=9.81, L=1: ω_0 ≈ 3.13 rad/s, Period ≈ 2 s **Damping Regimes:** Damping ratio: ζ = b/(2·ω_0) - ζ < 1: Underdamped (oscillatory decay) - ζ = 1: Critically damped (fastest non-oscillatory) - ζ > 1: Overdamped (slow exponential decay) **Effective Temperature:** T_eff = σ²/b Analogy with thermodynamics: - Noise injects energy: ~ σ² - Damping removes energy: ~ b - Equilibrium: Balance at T_eff **Energy Barrier:** From stable (θ=0) to unstable (θ=π): ΔE = 2·g/L For g=9.81, L=1: ΔE = 19.62 J/kg **Kramers Time:** Mean escape time: τ_escape ~ (b/ω_0²)·exp(ΔE·b/σ²) Example: b=0.5, σ=0.5, ΔE≈20 → τ ~ 0.05·exp(20) ~ 2.4×10⁸ seconds! **Noise Level Guidelines:** Low noise (σ < 0.3): - Rare escapes (τ > hours) - Nearly deterministic - Small fluctuations around equilibrium Moderate noise (σ = 0.3-1.0): - Occasional escapes (τ ~ minutes to hours) - Visible stochastic effects - Coherence resonance regime High noise (σ > 1.0): - Frequent escapes (τ ~ seconds) - Dominated by randomness - Large fluctuations **Stochastic Resonance Regime:** For periodic forcing u = A·cos(ω·t): - Optimal σ ≈ √(ΔE·b) ≈ √(2·g/L·b) - Maximizes response to weak signal ### estimate_kramers_escape_time { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.estimate_kramers_escape_time } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.estimate_kramers_escape_time( ) ``` Estimate mean escape time using Kramers formula. τ ~ (2πb/ω_0²)·exp(ΔE·b/σ²) #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|--------------------------------| | | float | Estimated mean escape time [s] | #### Notes {.doc-section .doc-section-notes} This is approximate, valid for: - Small noise: σ² << ΔE·b - Moderate damping: b ~ ω_0 For accurate times, use Monte Carlo simulation. #### Examples {.doc-section .doc-section-examples} ```python >>> pend = ContinuousStochasticPendulum(b=0.5, sigma=0.5) >>> tau = pend.estimate_kramers_escape_time() >>> print(f"Mean escape time: {tau:.2e} s") ``` ### get_effective_temperature { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_effective_temperature } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_effective_temperature( ) ``` Get effective temperature T_eff = σ²/b. Ratio of noise injection to dissipation. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-----------------------| | | float | Effective temperature | #### Notes {.doc-section .doc-section-notes} Analogy with thermodynamics: - Higher T_eff: More energetic fluctuations - Lower T_eff: Smaller fluctuations - Controls stationary distribution width #### Examples {.doc-section .doc-section-examples} ```python >>> pend = ContinuousStochasticPendulum(b=0.5, sigma=0.5) >>> T_eff = pend.get_effective_temperature() >>> print(f"T_eff: {T_eff:.3f}") ``` ### get_natural_frequency { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_natural_frequency } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.get_natural_frequency( ) ``` Get natural frequency ω_0 = √(g/L). #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------------| | | float | Natural frequency [rad/s] | #### Examples {.doc-section .doc-section-examples} ```python >>> pend = ContinuousStochasticPendulum(g=9.81, L=1.0) >>> omega_0 = pend.get_natural_frequency() >>> period = 2 * np.pi / omega_0 >>> print(f"Period: {period:.2f} s") ``` ### setup_equilibria { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.setup_equilibria } ```python systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.setup_equilibria( ) ``` Set up equilibrium points (deterministic part). Pendulum has periodic equilibria at θ = n·π.