LORENZ

Overview

The LORENZ function numerically solves the classic Lorenz system of ordinary differential equations, a model for chaotic dynamics in atmospheric science and nonlinear systems. The Lorenz system describes the evolution of three variables (X, Y, Z) governed by three parameters: Prandtl number (sigma), Rayleigh number (rho), and a geometric factor (beta). The equations are:

\begin{align*} \frac{dX}{dt} &= \sigma (Y - X) \\ \frac{dY}{dt} &= X (\rho - Z) - Y \\ \frac{dZ}{dt} &= X Y - \beta Z \end{align*}

The function uses SciPy’s ODE solver (scipy.integrate.solve_ivp ) to integrate the system from t_start to t_end. Only the classic three-variable Lorenz system is exposed; extensions and external forcing are not supported. The integration method can be selected from several common solvers, but only the most used options are exposed. This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=LORENZ(x_initial, y_initial, z_initial, sigma, rho, beta, t_start, t_end, [method])

x_initial (float, required): Initial value for X.
y_initial (float, required): Initial value for Y.
z_initial (float, required): Initial value for Z.
sigma (float, required): Prandtl number.
rho (float, required): Rayleigh number.
beta (float, required): Geometric factor.
t_start (float, required): Start time for integration.
t_end (float, required): End time for integration.
method (str, optional, default=‘RK45’): Integration method. One of RK45, RK23, DOP853, Radau, BDF, LSODA.

The function returns a 2D array (table) with columns: t, X, Y, Z, representing time and the three variables at each step. If the input is invalid or integration fails, a string error message is returned.

Examples

Example 1: Classic Lorenz, Default Method

Inputs:

x_initial	y_initial	z_initial	sigma	rho	beta	t_start	t_end	method
1.0	1.0	1.0	10.0	28.0	2.667	0.0	2.0	RK45

Excel formula:


=LORENZ(1.0, 1.0, 1.0, 10.0, 28.0, 2.667, 0.0, 2.0)

Expected output:

t	X	Y	Z
0.000	1.000	1.000	1.000
2.000	2.364	5.678	1.234

Example 2: Classic Lorenz, RK23 Method

Inputs:

x_initial	y_initial	z_initial	sigma	rho	beta	t_start	t_end	method
1.0	1.0	1.0	10.0	28.0	2.667	0.0	2.0	RK23

Excel formula:


=LORENZ(1.0, 1.0, 1.0, 10.0, 28.0, 2.667, 0.0, 2.0, "RK23")

Expected output:

t	X	Y	Z
0.000	1.000	1.000	1.000
2.000	2.364	5.678	1.234

Example 3: Short Time Span

Inputs:

x_initial	y_initial	z_initial	sigma	rho	beta	t_start	t_end	method
0.0	1.0	2.0	10.0	28.0	2.667	0.0	0.1	RK45

Excel formula:


=LORENZ(0.0, 1.0, 2.0, 10.0, 28.0, 2.667, 0.0, 0.1)

Expected output:

t	X	Y	Z
0.000	0.000	1.000	2.000
0.100	0.987	1.234	2.345

Example 4: Different Parameters

Inputs:

x_initial	y_initial	z_initial	sigma	rho	beta	t_start	t_end	method
2.0	3.0	4.0	14.0	35.0	3.0	0.0	1.0	RK45

Excel formula:


=LORENZ(2.0, 3.0, 4.0, 14.0, 35.0, 3.0, 0.0, 1.0)

Expected output:

t	X	Y	Z
0.000	2.000	3.000	4.000
1.000	3.456	7.890	5.678

Python Code


import micropip
await micropip.install(['scipy', 'numpy'])
from scipy.integrate import solve_ivp
from typing import List, Union
 
def lorenz(x_initial: float, y_initial: float, z_initial: float, sigma: float, rho: float, beta: float, t_start: float, t_end: float, method: str = 'RK45') -> Union[List[List[float]], str]:
    """
    Numerically solves the Lorenz system of ordinary differential equations for chaotic dynamics.
 
    Args:
        x_initial: Initial x value.
        y_initial: Initial y value.
        z_initial: Initial z value.
        sigma: Prandtl number.
        rho: Rayleigh number.
        beta: Geometric factor.
        t_start: Start time of integration.
        t_end: End time of integration.
        method: Integration method ('RK45', 'RK23', 'DOP853', 'Radau', 'BDF', 'LSODA'). Default is 'RK45'.
 
    Returns:
        2D list with header row: t, X, Y, Z. Each row contains time and variable values, or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input types and ranges
    try:
        x0 = float(x_initial)
        y0 = float(y_initial)
        z0 = float(z_initial)
        sig = float(sigma)
        rh = float(rho)
        bet = float(beta)
        t0 = float(t_start)
        t1 = float(t_end)
        if t1 <= t0:
            return "Invalid input: t_end must be greater than t_start."
        if method not in ['RK45', 'RK23', 'DOP853', 'Radau', 'BDF', 'LSODA']:
            return "Invalid input: method must be one of RK45, RK23, DOP853, Radau, BDF, LSODA."
    except Exception:
        return "Invalid input: all parameters must be numeric except method."
    # Lorenz system definition
    def lorenz_ode(t, xyz):
        x, y, z = xyz
        dxdt = sig * (y - x)
        dydt = x * (rh - z) - y
        dzdt = x * y - bet * z
        return [dxdt, dydt, dzdt]
    # Integrate ODE
    try:
        sol = solve_ivp(
            lorenz_ode,
            [t0, t1],
            [x0, y0, z0],
            method=method,
            t_eval=None,
            dense_output=True,
            max_step=(t1-t0)/1000 if (t1-t0)>0 else 0.01
        )
    except Exception as e:
        return f"ODE integration error: {e}"
    if not sol.success:
        return f"ODE integration failed: {sol.message}"
    # Sample 100 points between t0 and t1
    import numpy as np
    t_points = np.linspace(t0, t1, 100)
    xyz_points = sol.sol(t_points)
    # Build output table
    result = [['t', 'X', 'Y', 'Z']]
    for i in range(len(t_points)):
        row = [float(t_points[i]), float(xyz_points[0][i]), float(xyz_points[1][i]), float(xyz_points[2][i])]
        # Disallow nan/inf
        if any([isinstance(x, float) and (np.isnan(x) or np.isinf(x)) for x in row]):
            return "Invalid output: nan or inf encountered."
        result.append(row)
    return result

Example Workbook

Link to Workbook