# -*- coding: utf-8 -*-
import numpy as np
__all__ = [
'get_1d_stability_types',
'get_2d_stability_types',
'get_3d_stability_types',
'plot_scheme',
'stability_analysis',
'CENTER_MANIFOLD',
'SADDLE_NODE',
'STABLE_POINT_1D',
'UNSTABLE_POINT_1D',
'CENTER_2D',
'STABLE_NODE_2D',
'STABLE_FOCUS_2D',
'STABLE_STAR_2D',
'STABLE_DEGENERATE_2D',
'UNSTABLE_NODE_2D',
'UNSTABLE_FOCUS_2D',
'UNSTABLE_STAR_2D',
'UNSTABLE_DEGENERATE_2D',
'UNSTABLE_LINE_2D',
]
plot_scheme = {}
SADDLE_NODE = 'saddle node'
CENTER_MANIFOLD = 'center manifold'
plot_scheme[CENTER_MANIFOLD] = {'color': 'orangered'}
plot_scheme[SADDLE_NODE] = {"color": 'tab:blue'}
STABLE_POINT_1D = 'stable point'
UNSTABLE_POINT_1D = 'unstable point'
plot_scheme[STABLE_POINT_1D] = {"color": 'tab:red'}
plot_scheme[UNSTABLE_POINT_1D] = {"color": 'tab:olive'}
CENTER_2D = 'center'
STABLE_NODE_2D = 'stable node'
STABLE_FOCUS_2D = 'stable focus'
STABLE_STAR_2D = 'stable star'
STABLE_DEGENERATE_2D = 'stable degenerate'
UNSTABLE_NODE_2D = 'unstable node'
UNSTABLE_FOCUS_2D = 'unstable focus'
UNSTABLE_STAR_2D = 'unstable star'
UNSTABLE_DEGENERATE_2D = 'unstable degenerate'
UNSTABLE_LINE_2D = 'unstable line'
plot_scheme.update({
CENTER_2D: {'color': 'lime'},
STABLE_NODE_2D: {"color": 'tab:red'},
STABLE_FOCUS_2D: {"color": 'tab:purple'},
STABLE_STAR_2D: {'color': 'tab:olive'},
STABLE_DEGENERATE_2D: {'color': 'blueviolet'},
UNSTABLE_NODE_2D: {"color": 'tab:orange'},
UNSTABLE_FOCUS_2D: {"color": 'tab:cyan'},
UNSTABLE_STAR_2D: {'color': 'green'},
UNSTABLE_DEGENERATE_2D: {'color': 'springgreen'},
UNSTABLE_LINE_2D: {'color': 'dodgerblue'},
})
STABLE_POINT_3D = 'unclassified stable point'
UNSTABLE_POINT_3D = 'unclassified unstable point'
STABLE_NODE_3D = 'stable node'
UNSTABLE_SADDLE_3D = 'unstable saddle'
UNSTABLE_NODE_3D = 'unstable node'
STABLE_FOCUS_3D = 'stable focus'
UNSTABLE_FOCUS_3D = 'unstable focus'
UNSTABLE_CENTER_3D = 'unstable center'
UNKNOWN_3D = 'unknown 3d'
plot_scheme.update({
STABLE_POINT_3D: {'color': 'tab:gray'},
UNSTABLE_POINT_3D: {'color': 'tab:purple'},
STABLE_NODE_3D: {'color': 'tab:green'},
UNSTABLE_SADDLE_3D: {'color': 'tab:red'},
UNSTABLE_FOCUS_3D: {'color': 'tab:pink'},
STABLE_FOCUS_3D: {'color': 'tab:purple'},
UNSTABLE_NODE_3D: {'color': 'tab:orange'},
UNSTABLE_CENTER_3D: {'color': 'tab:olive'},
UNKNOWN_3D: {'color': 'tab:cyan'},
})
[docs]def get_1d_stability_types():
"""Get the stability types of 1D system."""
return [SADDLE_NODE, STABLE_POINT_1D, UNSTABLE_POINT_1D]
[docs]def get_2d_stability_types():
"""Get the stability types of 2D system."""
return [SADDLE_NODE, CENTER_2D, STABLE_NODE_2D, STABLE_FOCUS_2D,
STABLE_STAR_2D, CENTER_MANIFOLD, UNSTABLE_NODE_2D,
UNSTABLE_FOCUS_2D, UNSTABLE_STAR_2D, UNSTABLE_LINE_2D,
STABLE_DEGENERATE_2D, UNSTABLE_DEGENERATE_2D]
[docs]def get_3d_stability_types():
"""Get the stability types of 3D system."""
return [STABLE_POINT_3D, UNSTABLE_POINT_3D, STABLE_NODE_3D,
UNSTABLE_SADDLE_3D, UNSTABLE_NODE_3D, SADDLE_NODE,
STABLE_FOCUS_3D, UNSTABLE_FOCUS_3D, UNSTABLE_CENTER_3D, UNKNOWN_3D]
[docs]def stability_analysis(derivatives):
"""Stability analysis of fixed points for low-dimensional system.
The analysis is referred to [1]_.
Parameters
----------
derivatives : float, tuple, list, np.ndarray
The derivative of the f.
Returns
-------
fp_type : str
The type of the fixed point.
References
----------
.. [1] http://www.egwald.ca/nonlineardynamics/twodimensionaldynamics.php
"""
if np.size(derivatives) == 1: # 1D dynamical system
if derivatives == 0:
return SADDLE_NODE
elif derivatives > 0:
return UNSTABLE_POINT_1D
else:
return STABLE_POINT_1D
elif np.size(derivatives) == 4: # 2D dynamical system
a = derivatives[0][0]
b = derivatives[0][1]
c = derivatives[1][0]
d = derivatives[1][1]
# trace
p = a + d
# det
q = a * d - b * c
# judgement
if q < 0:
return SADDLE_NODE
elif q == 0:
if p <= 0:
return CENTER_MANIFOLD
else:
return UNSTABLE_LINE_2D
else:
# parabola
e = p * p - 4 * q
if p == 0:
return CENTER_2D
elif p > 0:
if e < 0:
return UNSTABLE_FOCUS_2D
elif e > 0:
return UNSTABLE_NODE_2D
else:
w = np.linalg.eigvals(derivatives)
if w[0] == w[1]:
return UNSTABLE_DEGENERATE_2D
else:
return UNSTABLE_STAR_2D
else:
if e < 0:
return STABLE_FOCUS_2D
elif e > 0:
return STABLE_NODE_2D
else:
w = np.linalg.eigvals(derivatives)
if w[0] == w[1]:
return STABLE_DEGENERATE_2D
else:
return STABLE_STAR_2D
elif np.size(derivatives) == 9: # 3D dynamical system
eigenvalues = np.linalg.eigvals(np.array(derivatives))
is_real = np.isreal(eigenvalues)
if is_real.all():
eigenvalues = np.sort(eigenvalues)
if eigenvalues[2] < 0:
return STABLE_NODE_3D
elif eigenvalues[2] == 0:
return UNKNOWN_3D
else:
if eigenvalues[0] > 0:
return UNSTABLE_NODE_3D
elif eigenvalues[0] == 0:
return UNKNOWN_3D
else:
if eigenvalues[1] < 0:
return SADDLE_NODE
elif eigenvalues[1] == 0:
return UNKNOWN_3D
else:
return UNSTABLE_SADDLE_3D
else:
if is_real.sum() == 1:
real_id = np.where(is_real)[0]
non_real_id = np.where(np.logical_not(is_real))[0]
v0 = eigenvalues[real_id]
v1 = eigenvalues[non_real_id[0]]
v2 = eigenvalues[non_real_id[1]]
v1_real = np.real(v1)
assert np.conj(v1) == v2
if v0 < 0:
if v1_real < 0:
return STABLE_FOCUS_3D
elif v1_real == 0: # 零实部
return UNKNOWN_3D
else:
return UNSTABLE_FOCUS_3D
elif v0 == 0:
if v1_real <= 0:
return UNKNOWN_3D # 零实部
else:
return UNSTABLE_POINT_3D # TODO
else:
if v1_real < 0:
return UNSTABLE_FOCUS_3D
elif v1_real == 0:
return UNSTABLE_CENTER_3D
else:
return UNSTABLE_POINT_3D # TODO
# else:
# raise ValueError()
eigenvalues = np.real(eigenvalues)
if np.all(eigenvalues < 0):
return STABLE_POINT_3D # TODO
else:
return UNSTABLE_POINT_3D # TODO
else:
raise ValueError('Unknown derivatives, only supports the jacobian '
'matrix with the shape of (1), (2, 2), or (3, 3).')
