Example1_long.py

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import numpy as np
import matplotlib.pyplot as plt
from pyhamsys import HamSys, Parameters

## Parameters
epsilon = 0.027						# parameter of the Hamiltonian system
N = 50  							# number of trajectories
nf = 1000							# number of points on the Poincaré section per trajectory

params = Parameters(				# integration parameters
	step = 2 * np.pi / 50,			# integration step size
	extension = True,				# use extended phase space integration
	check_energy = True,			# check energy conservation
	projection = 'symmetric'	  	# projection method used for extended phase space split integration
	)  

## Hamiltonian system and equations of motion 
def y_dot(t, y):
	x, p = np.split(y, 2)
	return np.concatenate((p, epsilon * (np.sin(x - t) + np.sin(x))), axis=None)

def k_dot(t, y):
	x = np.split(y, 2)[0]
	return -epsilon * np.sum(np.sin(x - t))

def hamiltonian(t, y):
	x, p = np.split(y, 2)
	return np.sum(p**2 / 2 + epsilon * (np.cos(x - t) + np.cos(x)))

hs = HamSys(ndof=N + 0.5, y_dot=y_dot, k_dot=k_dot, hamiltonian=hamiltonian)
	
## Initial conditions
x0 = 2 * np.pi * np.random.random(N)
p0 = np.random.random(N)
y0 = np.concatenate((x0, p0), axis=None)

## Integration
sol = hs.integrate(y0, 2 * np.pi * np.arange(nf + 1), params=params)

## Plot of the Poincaré section
fig, ax = plt.subplots()
x, p = np.split(sol.y, 2)
ax.plot(x % (2 * np.pi), p, '.b')
ax.set_xlim((0, 2 * np.pi))
plt.show()