pykoopman.common package

Submodules

pykoopman.common.cqgle module

Module for cubic-quintic Ginzburg-Landau equation.

class pykoopman.common.cqgle.cqgle(n, x, dt, tau=0.08, kappa=0, beta=0.66, nu=-0.1, sigma=-0.1, gamma=-0.1, L=6.283185307179586)

Bases: object

Cubic-quintic Ginzburg-Landau equation solver.

Solves the equation: i*u_t + (0.5 - i * tau) u_{xx} - i * kappa u_{xxxx} + (1-i * beta)|u|^2 u + (nu - i * sigma)|u|^4 u - i * gamma u = 0

Solves the periodic boundary conditions PDE using spectral methods.

Attributes:

• n_states (int) – Number of states.

• x (numpy.ndarray) – x-coordinates.

• dt (float) – Time step.

• tau (float) – Parameter tau.

• kappa (float) – Parameter kappa.

• beta (float) – Parameter beta.

• nu (float) – Parameter nu.

• sigma (float) – Parameter sigma.

• gamma (float) – Parameter gamma.

• k (numpy.ndarray) – Wave numbers.

• dk (float) – Wave number spacing.

sys(t, x, u)

System dynamics function.

simulate(x0, n_int, n_sample)

Simulate the system for a given initial condition.

collect_data_continuous(x0)

Collect training data pairs in continuous sense.

collect_one_step_data_discrete(x0)

Collect training data pairs in discrete sense.

collect_one_trajectory_data(x0, n_int, n_sample)

Collect data for one trajectory.

visualize_data(x, t, X)

Visualize the data in physical space.

visualize_state_space(X)

Visualize the data in state space.

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

pykoopman.common.examples module

module for example dynamics data

pykoopman.common.examples.drss(n=2, p=2, m=2, p_int_first=0.1, p_int_others=0.01, p_repeat=0.05, p_complex=0.5)

Create a discrete-time, random, stable, linear state space model.

Parameters:

• n (int, optional) – Number of states. Default is 2.

• p (int, optional) – Number of control inputs. Default is 2.

• m (int, optional) – Number of output measurements. If m=0, C becomes the identity matrix, so that y=x. Default is 2.

• p_int_first (float, optional) – Probability of an integrator as the first pole. Default is 0.1.

• p_int_others (float, optional) – Probability of other integrators beyond the first. Default is 0.01.

• p_repeat (float, optional) – Probability of repeated roots. Default is 0.05.

• p_complex (float, optional) – Probability of complex roots. Default is 0.5.

Returns:

A tuple containing the state transition matrix (A), control matrix (B), and measurement matrix (C).

A (numpy.ndarray): State transition matrix of shape (n, n). B (numpy.ndarray): Control matrix of shape (n, p). C (numpy.ndarray): Measurement matrix of shape (m, n). If m = 0, C is the

identity matrix.

Return type:

Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]

pykoopman.common.examples.advance_linear_system(x0, u, n, A=None, B=None, C=None)

Simulate the linear system dynamics for a given number of steps.

Parameters:

• x0 (numpy.ndarray) – Initial state vector of shape (n,).

• u (numpy.ndarray) – Control input array of shape (p,) or (p, n-1). If 1-dimensional, it will be converted to a row vector.

• n (int) – Number of steps to simulate.

• A (numpy.ndarray, optional) – State transition matrix of shape (n, n). If not provided, it defaults to None.

• B (numpy.ndarray, optional) – Control matrix of shape (n, p). If not provided, it defaults to None.

• C (numpy.ndarray, optional) – Measurement matrix of shape (m, n). If not provided, it defaults to None.

Returns:

A tuple containing the state trajectory (x) and the output trajectory (y).

x (numpy.ndarray): State trajectory of shape (n, len(x0)). y (numpy.ndarray): Output trajectory of shape (n, C.shape[0]).

Return type:

Tuple[numpy.ndarray, numpy.ndarray]

pykoopman.common.examples.vdp_osc(t, x, u)

Compute the dynamics of the Van der Pol oscillator.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – State vector of shape (2,).

• u (float) – Control input.

Returns:

Updated state vector of shape (2,).

Return type:

numpy.ndarray

pykoopman.common.examples.rk4(t, x, u, _dt=0.01, func=<function vdp_osc>)

Perform a 4th order Runge-Kutta integration.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – State vector of shape (2,).

• u (float) – Control input.

• _dt (float, optional) – Time step. Defaults to 0.01.

• func (function, optional) – Function defining the dynamics. Defaults to vdp_osc.

Returns:

Updated state vector of shape (2,).

Return type:

numpy.ndarray

pykoopman.common.examples.square_wave(step)

Generate a square wave with a period of 60 time steps.

Parameters:

step (int) – Current time step.

Returns:

Square wave value at the given time step.

Return type:

float

pykoopman.common.examples.sine_wave(step)

Generate a sine wave with a period of 60 time steps.

Parameters:

step (int) – Current time step.

Returns:

Sine wave value at the given time step.

Return type:

float

pykoopman.common.examples.lorenz(x, t, sigma=10, beta=2.6666666666666665, rho=28)

Compute the derivative of the Lorenz system at a given state.

Parameters:

• x (list) – Current state of the Lorenz system [x, y, z].

• t (float) – Current time.

• sigma (float, optional) – Parameter sigma. Default is 10.

• beta (float, optional) – Parameter beta. Default is 8/3.

• rho (float, optional) – Parameter rho. Default is 28.

Returns:

Derivative of the Lorenz system [dx/dt, dy/dt, dz/dt].

Return type:

list

pykoopman.common.examples.rev_dvdp(t, x, u=0, dt=0.1)

Reverse dynamics of the Van der Pol oscillator.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – Current state of the system [x1, x2].

• u (float, optional) – Input. Default is 0.

• dt (float, optional) – Time step. Default is 0.1.

Returns:

Updated state of the system [x1’, x2’].

Return type:

numpy.ndarray

class pykoopman.common.examples.Linear2Ddynamics

Bases: object

Initializes a Linear2Ddynamics object.

linear_map(x)

Applies the linear mapping to the input state.

Parameters:

x (numpy.ndarray) – Input state.

Returns:

Resulting mapped state.

Return type:

numpy.ndarray

collect_data(x, n_int, n_traj)

Collects data by integrating the linear dynamics.

Parameters:

• x (numpy.ndarray) – Initial state.

• n_int (int) – Number of integration steps.

• n_traj (int) – Number of trajectories.

Returns:

Input data. numpy.ndarray: Output data.

Return type:

numpy.ndarray

visualize_modes(x, phi, eigvals, order=None)

Visualizes the modes of the linear dynamics.

Parameters:

• x (numpy.ndarray) – State data.

• phi (numpy.ndarray) – Eigenvectors.

• eigvals (numpy.ndarray) – Eigenvalues.

• order (list, optional) – Order of the modes to visualize. Default is None.

class pykoopman.common.examples.torus_dynamics(n_states=128, sparsity=5, freq_max=15, noisemag=0.0)

Bases: object

Initializes a torus_dynamics object.

Parameters:

• n_states (int, optional) – Number of states. Default is 128.

• sparsity (int, optional) – Degree of sparsity. Default is 5.

• freq_max (int, optional) – Maximum frequency. Default is 15.

• noisemag (float, optional) – Magnitude of noise. Default is 0.0.

setup()

Sets up the dynamics.

advance(n_samples, dt=1)

Advances the continuous-time dynamics without control.

Parameters:

• n_samples (int) – Number of samples to advance.

• dt (float, optional) – Time step. Default is 1.

advance_discrete_time(n_samples, dt, u=None)

Advances the discrete-time dynamics with or without control.

Parameters:

• n_samples (int) – Number of samples to advance.

• dt (float) – Time step.

• u (array-like, optional) – Control input. Default is None.

set_control_matrix_physical(B)

Sets the control matrix in physical space.

Parameters:

B (array-like) – Control matrix in physical space.

set_control_matrix_fourier(Bhat)

Sets the control matrix in Fourier space.

Parameters:

Bhat (array-like) – Control matrix in Fourier space.

set_point_actuator(position=None)

Sets a single point actuator.

Parameters:

position (array-like, optional) – Position of the actuator. Default is None.

viz_setup()

Sets up the visualization.

viz_torus(ax, x)

Visualizes the torus dynamics.

Parameters:

• ax – Axes object for plotting.

• x (array-like) – Dynamics to be visualized.

Returns:

Surface plot of the torus dynamics.

Return type:

surface

viz_all_modes(modes=None)

Visualizes all modes.

Parameters:

modes (array-like, optional) – Modes to be visualized. Default is None.

Returns:

Figure object containing the visualizations.

Return type:

fig

property modes

Returns the modes of the dynamics.

Returns:

Modes of the dynamics.

Return type:

modes (array-like)

property B_effective

Returns the effective control matrix.

Returns:

Effective control matrix.

Return type:

B_effective (array-like)

class pykoopman.common.examples.slow_manifold(mu=-0.05, la=-1.0, dt=0.01)

Bases: object

Represents the slow manifold class.

Parameters:

• mu (float, optional) – Parameter mu. Default is -0.05.

• la (float, optional) – Parameter la. Default is -1.0.

• dt (float, optional) – Time step size. Default is 0.01.

Attributes:

• mu (float) – Parameter mu.

• la (float) – Parameter la.

• b (float) – Value computed from mu and la.

• dt (float) – Time step size.

• n_states (int) – Number of states.

sys(t, x, u)

Computes the system dynamics.

output(x)

Computes the output based on the state.

simulate(x0, n_int)

Simulates the system dynamics.

collect_data_continuous(x0)

Collects data from continuous-time dynamics.

collect_data_discrete(x0, n_int)

Collects data from discrete-time dynamics.

visualize_trajectories(t, X, n_traj)

Visualizes the trajectories.

visualize_state_space(X, Y, n_traj)

Visualizes the state space.

sys(t, x, u)

Computes the system dynamics.

Parameters:

• t (float) – Time.

• x (array-like) – State.

• u (array-like) – Control input.

Returns:

Computed system dynamics.

Return type:

array-like

output(x)

Computes the output based on the state.

Parameters:

x (array-like) – State.

Returns:

Computed output.

Return type:

array-like

simulate(x0, n_int)

Simulates the system dynamics.

Parameters:

• x0 (array-like) – Initial state.

• n_int (int) – Number of integration steps.

Returns:

Simulated trajectory.

Return type:

array-like

collect_data_continuous(x0)

Collects data from continuous-time dynamics.

Parameters:

x0 (array-like) – Initial state.

Returns:

Collected data (X, Y).

Return type:

tuple

collect_data_discrete(x0, n_int)

Collects data from discrete-time dynamics.

Parameters:

• x0 (array-like) – Initial state.

• n_int (int) – Number of integration steps.

Returns:

Collected data (X, Y).

Return type:

tuple

visualize_trajectories(t, X, n_traj)

Visualizes the trajectories.

Parameters:

• t (array-like) – Time vector.

• X (array-like) – State trajectories.

• n_traj (int) – Number of trajectories.

visualize_state_space(X, Y, n_traj)

Visualizes the state space.

Parameters:

• X (array-like) – State trajectories.

• Y (array-like) – Output trajectories.

• n_traj (int) – Number of trajectories.

class pykoopman.common.examples.forced_duffing(dt, d, alpha, beta)

Bases: object

Initializes the Forced Duffing Oscillator.

Parameters:

• dt (float) – Time step.

• d (float) – Damping coefficient.

• alpha (float) – Coefficient of x1.

• beta (float) – Coefficient of x1^3.

sys(t, x, u)

Defines the system dynamics of the Forced Duffing Oscillator.

Parameters:

• t (float) – Time.

• x (array-like) – State vector.

• u (array-like) – Control input.

Returns:

Rate of change of the state vector.

Return type:

array-like

simulate(x0, n_int, u)

Simulates the Forced Duffing Oscillator.

Parameters:

• x0 (array-like) – Initial state vector.

• n_int (int) – Number of time steps.

• u (array-like) – Control inputs.

Returns:

State trajectories.

Return type:

array-like

collect_data_continuous(x0, u)

Collects continuous data for the Forced Duffing Oscillator.

Parameters:

• x0 (array-like) – Initial state vector.

• u (array-like) – Control inputs.

Returns:

State and output trajectories.

Return type:

tuple

collect_data_discrete(x0, n_int, u)

Collects discrete-time data for the Forced Duffing Oscillator.

Parameters:

• x0 (array-like) – Initial state vector.

• n_int (int) – Number of time steps.

• u (array-like) – Control inputs.

Returns:

State and output trajectories.

Return type:

tuple

visualize_trajectories(t, X, n_traj)

Visualizes the state trajectories of the Forced Duffing Oscillator.

Parameters:

• t (array-like) – Time vector.

• X (array-like) – State trajectories.

• n_traj (int) – Number of trajectories to visualize.

visualize_state_space(X, Y, n_traj)

Visualizes the state space trajectories of the Forced Duffing Oscillator.

Parameters:

• X (array-like) – State trajectories.

• Y (array-like) – Output trajectories.

• n_traj (int) – Number of trajectories to visualize.

pykoopman.common.ks module

module for 1D KS equation

class pykoopman.common.ks.ks(n, x, nu, dt, M=16)

Bases: object

Solving 1D KS equation

u_t = -u*u_x + u_{xx} + nu*u_{xxxx}

Periodic B.C. between 0 and 2*pi. This PDE is solved using spectral methods.

static compute_u2k_zeropad_dealiased(uk_)

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

pykoopman.common.nlse module

module for nonlinear schrodinger equation

class pykoopman.common.nlse.nlse(n, dt, L=6.283185307179586)

Bases: object

nonlinear schrodinger equation

iu_t + 0.5u_xx + u*|u|^2 = 0

periodic B.C. PDE is solved with Spectral methods using FFT

sys(t, x, u)

the RHS for the governing equation using FFT

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

pykoopman.common.validation module

pykoopman.common.validation.validate_input(x, t=<object object>)

pykoopman.common.validation.check_array(x, **kwargs)

pykoopman.common.validation.drop_nan_rows(arr, *args)

Remove rows in all inputs for which arr has _np.nan entries.

Parameters:

• arr (numpy.ndarray) – Array whose rows are checked for nan entries. Any rows containing nans are removed from arr and all arguments passed via args.

• *args (variable length argument list of numpy.ndarray) – Additional arrays from which to remove rows. Each argument should have the same number of rows as arr.

Returns:

arrays – Arrays with nan rows dropped. The first entry corresponds to arr and all following entries to *args.

Return type:

tuple of numpy.ndarray

pykoopman.common.vbe module

module for 1D viscous burgers

class pykoopman.common.vbe.vbe(n, x, dt, nu=0.1, L=6.283185307179586)

Bases: object

1D viscous Burgers equation

u_t = -u*u_x +

u u_{xx}

periodic B.C. PDE is solved using spectral methods

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

Module contents

pykoopman.common.check_array(x, **kwargs)

pykoopman.common.drop_nan_rows(arr, *args)

Remove rows in all inputs for which arr has _np.nan entries.

Parameters:

• arr (numpy.ndarray) – Array whose rows are checked for nan entries. Any rows containing nans are removed from arr and all arguments passed via args.

• *args (variable length argument list of numpy.ndarray) – Additional arrays from which to remove rows. Each argument should have the same number of rows as arr.

Returns:

arrays – Arrays with nan rows dropped. The first entry corresponds to arr and all following entries to *args.

Return type:

tuple of numpy.ndarray

pykoopman.common.validate_input(x, t=<object object>)

pykoopman.common.drss(n=2, p=2, m=2, p_int_first=0.1, p_int_others=0.01, p_repeat=0.05, p_complex=0.5)

Create a discrete-time, random, stable, linear state space model.

Parameters:

• n (int, optional) – Number of states. Default is 2.

• p (int, optional) – Number of control inputs. Default is 2.

• m (int, optional) – Number of output measurements. If m=0, C becomes the identity matrix, so that y=x. Default is 2.

• p_int_first (float, optional) – Probability of an integrator as the first pole. Default is 0.1.

• p_int_others (float, optional) – Probability of other integrators beyond the first. Default is 0.01.

• p_repeat (float, optional) – Probability of repeated roots. Default is 0.05.

• p_complex (float, optional) – Probability of complex roots. Default is 0.5.

Returns:

A tuple containing the state transition matrix (A), control matrix (B), and measurement matrix (C).

A (numpy.ndarray): State transition matrix of shape (n, n). B (numpy.ndarray): Control matrix of shape (n, p). C (numpy.ndarray): Measurement matrix of shape (m, n). If m = 0, C is the

identity matrix.

Return type:

Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]

pykoopman.common.advance_linear_system(x0, u, n, A=None, B=None, C=None)

Simulate the linear system dynamics for a given number of steps.

Parameters:

• x0 (numpy.ndarray) – Initial state vector of shape (n,).

• u (numpy.ndarray) – Control input array of shape (p,) or (p, n-1). If 1-dimensional, it will be converted to a row vector.

• n (int) – Number of steps to simulate.

• A (numpy.ndarray, optional) – State transition matrix of shape (n, n). If not provided, it defaults to None.

• B (numpy.ndarray, optional) – Control matrix of shape (n, p). If not provided, it defaults to None.

• C (numpy.ndarray, optional) – Measurement matrix of shape (m, n). If not provided, it defaults to None.

Returns:

A tuple containing the state trajectory (x) and the output trajectory (y).

x (numpy.ndarray): State trajectory of shape (n, len(x0)). y (numpy.ndarray): Output trajectory of shape (n, C.shape[0]).

Return type:

Tuple[numpy.ndarray, numpy.ndarray]

class pykoopman.common.torus_dynamics(n_states=128, sparsity=5, freq_max=15, noisemag=0.0)

Bases: object

Initializes a torus_dynamics object.

Parameters:

• n_states (int, optional) – Number of states. Default is 128.

• sparsity (int, optional) – Degree of sparsity. Default is 5.

• freq_max (int, optional) – Maximum frequency. Default is 15.

• noisemag (float, optional) – Magnitude of noise. Default is 0.0.

setup()

Sets up the dynamics.

advance(n_samples, dt=1)

Advances the continuous-time dynamics without control.

Parameters:

• n_samples (int) – Number of samples to advance.

• dt (float, optional) – Time step. Default is 1.

advance_discrete_time(n_samples, dt, u=None)

Advances the discrete-time dynamics with or without control.

Parameters:

• n_samples (int) – Number of samples to advance.

• dt (float) – Time step.

• u (array-like, optional) – Control input. Default is None.

set_control_matrix_physical(B)

Sets the control matrix in physical space.

Parameters:

B (array-like) – Control matrix in physical space.

set_control_matrix_fourier(Bhat)

Sets the control matrix in Fourier space.

Parameters:

Bhat (array-like) – Control matrix in Fourier space.

set_point_actuator(position=None)

Sets a single point actuator.

Parameters:

position (array-like, optional) – Position of the actuator. Default is None.

viz_setup()

Sets up the visualization.

viz_torus(ax, x)

Visualizes the torus dynamics.

Parameters:

• ax – Axes object for plotting.

• x (array-like) – Dynamics to be visualized.

Returns:

Surface plot of the torus dynamics.

Return type:

surface

viz_all_modes(modes=None)

Visualizes all modes.

Parameters:

modes (array-like, optional) – Modes to be visualized. Default is None.

Returns:

Figure object containing the visualizations.

Return type:

fig

property modes

Returns the modes of the dynamics.

Returns:

Modes of the dynamics.

Return type:

modes (array-like)

property B_effective

Returns the effective control matrix.

Returns:

Effective control matrix.

Return type:

B_effective (array-like)

pykoopman.common.lorenz(x, t, sigma=10, beta=2.6666666666666665, rho=28)

Compute the derivative of the Lorenz system at a given state.

Parameters:

• x (list) – Current state of the Lorenz system [x, y, z].

• t (float) – Current time.

• sigma (float, optional) – Parameter sigma. Default is 10.

• beta (float, optional) – Parameter beta. Default is 8/3.

• rho (float, optional) – Parameter rho. Default is 28.

Returns:

Derivative of the Lorenz system [dx/dt, dy/dt, dz/dt].

Return type:

list

pykoopman.common.vdp_osc(t, x, u)

Compute the dynamics of the Van der Pol oscillator.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – State vector of shape (2,).

• u (float) – Control input.

Returns:

Updated state vector of shape (2,).

Return type:

numpy.ndarray

pykoopman.common.rk4(t, x, u, _dt=0.01, func=<function vdp_osc>)

Perform a 4th order Runge-Kutta integration.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – State vector of shape (2,).

• u (float) – Control input.

• _dt (float, optional) – Time step. Defaults to 0.01.

• func (function, optional) – Function defining the dynamics. Defaults to vdp_osc.

Returns:

Updated state vector of shape (2,).

Return type:

numpy.ndarray

pykoopman.common.rev_dvdp(t, x, u=0, dt=0.1)

Reverse dynamics of the Van der Pol oscillator.

Parameters:

• t (float) – Time.

• x (numpy.ndarray) – Current state of the system [x1, x2].

• u (float, optional) – Input. Default is 0.

• dt (float, optional) – Time step. Default is 0.1.

Returns:

Updated state of the system [x1’, x2’].

Return type:

numpy.ndarray

class pykoopman.common.Linear2Ddynamics

Bases: object

Initializes a Linear2Ddynamics object.

linear_map(x)

Applies the linear mapping to the input state.

Parameters:

x (numpy.ndarray) – Input state.

Returns:

Resulting mapped state.

Return type:

numpy.ndarray

collect_data(x, n_int, n_traj)

Collects data by integrating the linear dynamics.

Parameters:

• x (numpy.ndarray) – Initial state.

• n_int (int) – Number of integration steps.

• n_traj (int) – Number of trajectories.

Returns:

Input data. numpy.ndarray: Output data.

Return type:

numpy.ndarray

visualize_modes(x, phi, eigvals, order=None)

Visualizes the modes of the linear dynamics.

Parameters:

• x (numpy.ndarray) – State data.

• phi (numpy.ndarray) – Eigenvectors.

• eigvals (numpy.ndarray) – Eigenvalues.

• order (list, optional) – Order of the modes to visualize. Default is None.

class pykoopman.common.slow_manifold(mu=-0.05, la=-1.0, dt=0.01)

Bases: object

Represents the slow manifold class.

Parameters:

• mu (float, optional) – Parameter mu. Default is -0.05.

• la (float, optional) – Parameter la. Default is -1.0.

• dt (float, optional) – Time step size. Default is 0.01.

Attributes:

• mu (float) – Parameter mu.

• la (float) – Parameter la.

• b (float) – Value computed from mu and la.

• dt (float) – Time step size.

• n_states (int) – Number of states.

sys(t, x, u)

Computes the system dynamics.

output(x)

Computes the output based on the state.

simulate(x0, n_int)

Simulates the system dynamics.

collect_data_continuous(x0)

Collects data from continuous-time dynamics.

collect_data_discrete(x0, n_int)

Collects data from discrete-time dynamics.

visualize_trajectories(t, X, n_traj)

Visualizes the trajectories.

visualize_state_space(X, Y, n_traj)

Visualizes the state space.

sys(t, x, u)

Computes the system dynamics.

Parameters:

• t (float) – Time.

• x (array-like) – State.

• u (array-like) – Control input.

Returns:

Computed system dynamics.

Return type:

array-like

output(x)

Computes the output based on the state.

Parameters:

x (array-like) – State.

Returns:

Computed output.

Return type:

array-like

simulate(x0, n_int)

Simulates the system dynamics.

Parameters:

• x0 (array-like) – Initial state.

• n_int (int) – Number of integration steps.

Returns:

Simulated trajectory.

Return type:

array-like

collect_data_continuous(x0)

Collects data from continuous-time dynamics.

Parameters:

x0 (array-like) – Initial state.

Returns:

Collected data (X, Y).

Return type:

tuple

collect_data_discrete(x0, n_int)

Collects data from discrete-time dynamics.

Parameters:

• x0 (array-like) – Initial state.

• n_int (int) – Number of integration steps.

Returns:

Collected data (X, Y).

Return type:

tuple

visualize_trajectories(t, X, n_traj)

Visualizes the trajectories.

Parameters:

• t (array-like) – Time vector.

• X (array-like) – State trajectories.

• n_traj (int) – Number of trajectories.

visualize_state_space(X, Y, n_traj)

Visualizes the state space.

Parameters:

• X (array-like) – State trajectories.

• Y (array-like) – Output trajectories.

• n_traj (int) – Number of trajectories.

class pykoopman.common.nlse(n, dt, L=6.283185307179586)

Bases: object

nonlinear schrodinger equation

iu_t + 0.5u_xx + u*|u|^2 = 0

periodic B.C. PDE is solved with Spectral methods using FFT

sys(t, x, u)

the RHS for the governing equation using FFT

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

class pykoopman.common.vbe(n, x, dt, nu=0.1, L=6.283185307179586)

Bases: object

1D viscous Burgers equation

u_t = -u*u_x +

u u_{xx}

periodic B.C. PDE is solved using spectral methods

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

class pykoopman.common.cqgle(n, x, dt, tau=0.08, kappa=0, beta=0.66, nu=-0.1, sigma=-0.1, gamma=-0.1, L=6.283185307179586)

Bases: object

Cubic-quintic Ginzburg-Landau equation solver.

Solves the equation: i*u_t + (0.5 - i * tau) u_{xx} - i * kappa u_{xxxx} + (1-i * beta)|u|^2 u + (nu - i * sigma)|u|^4 u - i * gamma u = 0

Solves the periodic boundary conditions PDE using spectral methods.

Attributes:

• n_states (int) – Number of states.

• x (numpy.ndarray) – x-coordinates.

• dt (float) – Time step.

• tau (float) – Parameter tau.

• kappa (float) – Parameter kappa.

• beta (float) – Parameter beta.

• nu (float) – Parameter nu.

• sigma (float) – Parameter sigma.

• gamma (float) – Parameter gamma.

• k (numpy.ndarray) – Wave numbers.

• dk (float) – Wave number spacing.

sys(t, x, u)

System dynamics function.

simulate(x0, n_int, n_sample)

Simulate the system for a given initial condition.

collect_data_continuous(x0)

Collect training data pairs in continuous sense.

collect_one_step_data_discrete(x0)

Collect training data pairs in discrete sense.

collect_one_trajectory_data(x0, n_int, n_sample)

Collect data for one trajectory.

visualize_data(x, t, X)

Visualize the data in physical space.

visualize_state_space(X)

Visualize the data in state space.

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect training data pairs - continuous sense.

given x0, with shape (n_dim, n_traj), the function returns dx/dt with shape (n_dim, n_traj)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)

class pykoopman.common.ks(n, x, nu, dt, M=16)

Bases: object

Solving 1D KS equation

u_t = -u*u_x + u_{xx} + nu*u_{xxxx}

Periodic B.C. between 0 and 2*pi. This PDE is solved using spectral methods.

static compute_u2k_zeropad_dealiased(uk_)

sys(t, x, u)

simulate(x0, n_int, n_sample)

collect_data_continuous(x0)

collect_one_step_data_discrete(x0)

collect training data pairs - discrete sense.

given x0, with shape (n_dim, n_traj), the function returns system state x1 after self.dt with shape (n_dim, n_traj)

collect_one_trajectory_data(x0, n_int, n_sample)

visualize_data(x, t, X)

visualize_state_space(X)