In this notebook we will show how to calculate Lagrangian descriptors for a 1 DoF dynamical system defined by the user.
We begin by importing the LDDS module, and from it, all the functions necessary to calculate and plot LDs.
import os, sys
sys.path.insert(1, os.pardir)
import ldds
from ldds.base import compute_lagrangian_descriptor
from ldds.tools import draw_all_lds
import numpy as np
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings('ignore')
The Hamiltonian function that describes the Morse oscillator is:
\begin{equation*} H(x,p_x) = \dfrac{1}{2m}p_x^2 + D\left(1 - e^{-\alpha x}\right)^2 \end{equation*}where $m$ is the mass, $D$ represents the well depth and $\alpha$ controls the well width. The vector field that defines the dynamical system is given by:
\begin{align} \dot{x} & = \dfrac{\partial H}{\partial p_x} = f_1(x,p_x) = \dfrac{p_x}{m} \\[.2cm] \dot{p}_x & = -\dfrac{\partial H}{\partial x} = f_2(x,p_x) = -2 D\,\alpha \left(e^{-\alpha x} - e^{-2\alpha x}\right) \end{align}Define the Morse1D function containing the two-dimensional dynamical system
def Morse1D(t, u, PARAMETERS = [1, 2, 1]):
"""
Returns a user-defined 1-DoF vector field at time t, for an array of points in phase-space.
Number of model parameters: 1 . PARAMETERS = [m, D, alpha]
Functional form: v = ((1/m)*y, -2*D*alpha*(exp(-alpha*x) - exp(-2*alpha*x))), with u = (x, y)
Parameters
----------
t : float
fixed time-point of vector field, for all points in phase-space.
u : array_like, shape(n,)
points in phase-space to determine vector field at common time t.
PARAMETERS : list of floats
vector field parameters
Returns
-------
v : array_like, shape(n,)
vector field corresponding to points u, in phase-space at time t
"""
x, y = u.T
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#
# CHANGE STUFF BELOW ONLY
#
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# Hamiltonian Model Parameter
m, D, alpha = PARAMETERS
v = np.column_stack([ (1/m) * y, -2 * D * alpha * (np.exp(-alpha * x) - np.exp(-2 * alpha * x))])
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#
# CHANGE STUFF ABOVE ONLY
#
##############################################
return v
# Mesh parameters to calculate LDs
# 2D Domain
x_min,x_max = [-1, 7]
y_min,y_max = [-2.5, 2.5]
# Grid size
Nx, Ny = [500, 500]
grid_parameters = [(x_min, x_max, Nx), (y_min, y_max, Ny)]
# Morse oscillator parameters
m, D, alpha = [1, 2, 1]
vector_field = lambda t,u: Morse1D(t, u, PARAMETERS = [m, D, alpha])
# Integration parameters (Calculate LDs at time t = t0 by integrating in the interval [t0-tau,t0+tau])
t0 = 0
tau = 15
# LDs definition (we will use the p-value seminorm)
p_value = 0.5
LD_forward = compute_lagrangian_descriptor(grid_parameters, vector_field, tau, p_value)
LD_backward = compute_lagrangian_descriptor(grid_parameters, vector_field, -tau, p_value)
figs = draw_all_lds(LD_forward, LD_backward, grid_parameters, tau, p_value)
We change next the model parameters:
# Morse oscillator parameters
m, D, alpha = [1, 1, 1]
vector_field = lambda t,u: Morse1D(t, u, PARAMETERS = [m, D, alpha])
# Integration parameters (Calculate LDs at time t = t0 by integrating in the interval [t0-tau,t0+tau])
t0 = 0
tau = 15
# LDs definition (we will use the arclength)
p_value = 2
LD_forward = compute_lagrangian_descriptor(grid_parameters, vector_field, tau, p_value)
LD_backward = compute_lagrangian_descriptor(grid_parameters, vector_field, -tau, p_value)
figs = draw_all_lds(LD_forward, LD_backward, grid_parameters, tau, p_value)
