generate

Functions:

Name	Description
double_pendulum	Generate double pendulum dynamics via forward Euler integration.
to_xy	Convert double pendulum angles to Cartesian coordinates.
lorenz	Generate a Lorenz system trajectory via forward Euler integration.
mackey_glass	Generate a Mackey-Glass chaotic time series via forward Euler integration.

double_pendulum

double_pendulum(m1: float, m2: float, L1: float, L2: float, g: float, X0: np.ndarray, dt: float, t_max: int)

Generate double pendulum dynamics via forward Euler integration.

Parameters:

Name	Type	Description	Default
m1	float	Mass of first pendulum.	required
m2	float	Mass of second pendulum.	required
L1	float	Length of first pendulum.	required
L2	float	Length of second pendulum.	required
g	float	Gravitational acceleration.	required
X0	ndarray	Initial state (theta1, theta2, omega1, omega2) of shape (4,).	required
dt	float	Integration time step.	required
t_max	int	Maximum time.	required

Returns:

Name	Type	Description
t	ndarray	Time array.
X	ndarray	State trajectory of shape (N, 4).

to_xy

to_xy(L1: float, L2: float, theta1: np.ndarray, theta2: np.ndarray)

Convert double pendulum angles to Cartesian coordinates.

Parameters:

Name	Type	Description	Default
L1	float	Length of first pendulum.	required
L2	float	Length of second pendulum.	required
theta1	ndarray	Angle of first pendulum.	required
theta2	ndarray	Angle of second pendulum.	required

Returns:

Name	Type	Description
x1	ndarray	x-coordinate of first pendulum.
y1	ndarray	y-coordinate of first pendulum.
x2	ndarray	x-coordinate of second pendulum.
y2	ndarray	y-coordinate of second pendulum.

lorenz

lorenz(sigma: float, rho: float, beta: float, X0: np.ndarray, dt: float, t_max: int)

Generate a Lorenz system trajectory via forward Euler integration.

Parameters:

Name	Type	Description	Default
sigma	float	Prandtl number (typical: 10).	required
rho	float	Rayleigh number (typical: 28).	required
beta	float	Geometric factor (typical: 8/3).	required
X0	ndarray	Initial condition of shape (3,).	required
dt	float	Integration time step.	required
t_max	int	Maximum time.	required

Returns:

Name	Type	Description
t	ndarray	Time array.
X	ndarray	Trajectory of shape (N, 3) for (x, y, z).

mackey_glass

mackey_glass(tau: float, n: int, beta: float, gamma: float, x0: float, dt: float, t_max: int)

Generate a Mackey-Glass chaotic time series via forward Euler integration.

Parameters:

Name	Type	Description	Default
tau	float	Delay parameter (typical: 17 for chaos).	required
n	int	Nonlinearity exponent (typical: 10).	required
beta	float	Feedback strength (typical: 0.2).	required
gamma	float	Decay rate (typical: 0.1).	required
x0	float	Initial condition.	required
dt	float	Integration time step.	required
t_max	int	Maximum time.	required

Returns:

Name	Type	Description
t	ndarray	Time array.
x	ndarray	1D time series.