Repository navigation
Expand file tree
/
Copy pathphaseplot.py
More file actions
1533 lines (1280 loc) · 58.3 KB
/
Copy pathphaseplot.py
File metadata and controls
1533 lines (1280 loc) · 58.3 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
# phaseplot.py - generate 2D phase portraits
#
# Initial author: Richard M. Murray
# Creation date: 24 July 2011, converted from MATLAB version (2002);
# based on an original version by Kristi Morgansen
"""Generate 2D phase portraits.
This module contains functions for generating 2D phase plots. The base
function for creating phase plane portraits is `~control.phase_plane_plot`,
which generates a phase plane portrait for a 2 state I/O system (with no
inputs). Utility functions are available to customize the individual
elements of a phase plane portrait.
The docstring examples assume the following import commands::
>>> import numpy as np
>>> import control as ct
>>> import control.phaseplot as pp
"""
import math
import warnings
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint
from . import config
from .ctrlplot import ControlPlot, _add_arrows_to_line2D, _get_color, \
_process_ax_keyword, _update_plot_title
from .exception import ControlArgument
from .nlsys import NonlinearIOSystem, find_operating_point, \
input_output_response
__all__ = ['phase_plane_plot', 'phase_plot', 'box_grid']
# Default values for module parameter variables
_phaseplot_defaults = {
'phaseplot.arrows': 2, # number of arrows around curve
'phaseplot.arrow_size': 8, # pixel size for arrows
'phaseplot.arrow_style': None, # set arrow style
'phaseplot.separatrices_radius': 0.1 # initial radius for separatrices
}
def phase_plane_plot(
sys, pointdata=None, timedata=None, gridtype=None, gridspec=None,
plot_streamlines=None, plot_vectorfield=None, plot_streamplot=None,
plot_equilpoints=True, plot_separatrices=True, ax=None,
suppress_warnings=False, title=None, **kwargs
):
"""Plot phase plane diagram.
This function plots phase plane data, including vector fields, stream
lines, equilibrium points, and contour curves.
If none of plot_streamlines, plot_vectorfield, or plot_streamplot are
set, then plot_streamplot is used by default.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int, 1D array, or 2D array
Time to simulate each streamline. If a 1D array is given, then the
times at which to sample the simulation for all streamlines.
If a 2D array is given, then for each row the times at which
to sample the simulation for the corresponding streamline.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot all elements in the given color (use ``plot_<element>`` =
{'color': c} to set the color in one element of the phase
plot (equilpoints, separatrices, streamlines, etc).
ax : `matplotlib.axes.Axes`, optional
The matplotlib axes to draw the figure on. If not specified and
the current figure has a single axes, that axes is used.
Otherwise, a new figure is created.
Returns
-------
cplt : `ControlPlot` object
Object containing the data that were plotted. See `ControlPlot`
for more detailed information.
cplt.lines : array of list of `matplotlib.lines.Line2D`
Array of list of `matplotlib.artist.Artist` objects:
- lines[0] = list of Line2D objects (streamlines, separatrices).
- lines[1] = Quiver object (vector field arrows).
- lines[2] = list of Line2D objects (equilibrium points).
- lines[3] = StreamplotSet object (lines with arrows).
cplt.axes : 2D array of `matplotlib.axes.Axes`
Axes for each subplot.
cplt.figure : `matplotlib.figure.Figure`
Figure containing the plot.
Other Parameters
----------------
arrows : int
Set the number of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrows']`.
arrow_size : float
Set the size of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_size']`.
arrow_style : matplotlib patch
Set the style of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_style']`.
dir : str, optional
Direction to draw streamlines: 'forward' to flow forward in time
from the reference points, 'reverse' to flow backward in time, or
'both' to flow both forward and backward. The amount of time to
simulate in each direction is given by the `timedata` argument.
plot_streamlines : bool or dict, optional
If True then plot streamlines based on the pointdata and gridtype.
If set to a dict, pass on the key-value pairs in the dict as
keywords to `streamlines`.
plot_vectorfield : bool or dict, optional
If True then plot the vector field based on the pointdata and
gridtype. If set to a dict, pass on the key-value pairs in the
dict as keywords to `phaseplot.vectorfield`.
plot_streamplot : bool or dict, optional
If True then use `matplotlib.axes.Axes.streamplot` function
to plot the streamlines. If set to a dict, pass on the key-value
pairs in the dict as keywords to `phaseplot.streamplot`.
plot_equilpoints : bool or dict, optional
If True (default) then plot equilibrium points based in the phase
plot boundary. If set to a dict, pass on the key-value pairs in the
dict as keywords to `phaseplot.equilpoints`.
plot_separatrices : bool or dict, optional
If True (default) then plot separatrices starting from each
equilibrium point. If set to a dict, pass on the key-value pairs
in the dict as keywords to `phaseplot.separatrices`.
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
title : str, optional
Set the title of the plot. Defaults to plot type and system name(s).
Notes
-----
The default method for producing streamlines is determined based on which
keywords are specified, with `plot_streamplot` serving as the generic
default. If any of the `arrows`, `arrow_size`, `arrow_style`, or `dir`
keywords are used and neither `plot_streamlines` nor `plot_streamplot` is
set, then `plot_streamlines` will be set to True. If neither
`plot_streamlines` nor `plot_vectorfield` set set to True, then
`plot_streamplot` will be set to True.
"""
# Check for legacy usage of plot_streamlines
streamline_keywords = [
'arrows', 'arrow_size', 'arrow_style', 'dir']
if plot_streamlines is None:
if any([kw in kwargs for kw in streamline_keywords]):
warnings.warn(
"detected streamline keywords; use plot_streamlines to set",
FutureWarning)
plot_streamlines = True
if gridtype not in [None, 'meshgrid']:
warnings.warn(
"streamplots only support gridtype='meshgrid'; "
"falling back to streamlines")
plot_streamlines = True
if plot_streamlines is None and plot_vectorfield is None \
and plot_streamplot is None:
plot_streamplot = True
if plot_streamplot and not plot_streamlines and not plot_vectorfield:
gridspec = gridspec or [25, 25]
# Process arguments
params = kwargs.get('params', None)
sys = _create_system(sys, params)
pointdata = [-1, 1, -1, 1] if pointdata is None else pointdata
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Create axis if needed
user_ax = ax
fig, ax = _process_ax_keyword(user_ax, squeeze=True, rcParams=rcParams)
# Create copy of kwargs for later checking to find unused arguments
initial_kwargs = dict(kwargs)
# Utility function to create keyword arguments
def _create_kwargs(global_kwargs, local_kwargs, **other_kwargs):
new_kwargs = dict(global_kwargs)
new_kwargs.update(other_kwargs)
if isinstance(local_kwargs, dict):
new_kwargs.update(local_kwargs)
return new_kwargs
# Create list for storing outputs
out = np.array([[], None, None, None], dtype=object)
# the maximum zorder of stramlines, vectorfield or streamplot
flow_zorder = None
# Plot out the main elements
if plot_streamlines:
kwargs_local = _create_kwargs(
kwargs, plot_streamlines, gridspec=gridspec, gridtype=gridtype,
ax=ax)
out[0] += streamlines(
sys, pointdata, timedata, _check_kwargs=False,
suppress_warnings=suppress_warnings, **kwargs_local)
new_zorder = max(elem.get_zorder() for elem in out[0])
flow_zorder = max(flow_zorder, new_zorder) if flow_zorder \
else new_zorder
# Get rid of keyword arguments handled by streamlines
for kw in ['arrows', 'arrow_size', 'arrow_style', 'color',
'dir', 'params']:
initial_kwargs.pop(kw, None)
# Reset the gridspec for the remaining commands, if needed
if gridtype not in [None, 'boxgrid', 'meshgrid']:
gridspec = None
if plot_vectorfield:
kwargs_local = _create_kwargs(
kwargs, plot_vectorfield, gridspec=gridspec, ax=ax)
out[1] = vectorfield(
sys, pointdata, _check_kwargs=False, **kwargs_local)
new_zorder = out[1].get_zorder()
flow_zorder = max(flow_zorder, new_zorder) if flow_zorder \
else new_zorder
# Get rid of keyword arguments handled by vectorfield
for kw in ['color', 'params']:
initial_kwargs.pop(kw, None)
if plot_streamplot:
if gridtype not in [None, 'meshgrid']:
raise ValueError(
"gridtype must be 'meshgrid' when using streamplot")
kwargs_local = _create_kwargs(
kwargs, plot_streamplot, gridspec=gridspec, ax=ax)
out[3] = streamplot(
sys, pointdata, _check_kwargs=False, **kwargs_local)
new_zorder = max(out[3].lines.get_zorder(), out[3].arrows.get_zorder())
flow_zorder = max(flow_zorder, new_zorder) if flow_zorder \
else new_zorder
# Get rid of keyword arguments handled by streamplot
for kw in ['color', 'params']:
initial_kwargs.pop(kw, None)
sep_zorder = flow_zorder + 1 if flow_zorder else None
if plot_separatrices:
kwargs_local = _create_kwargs(
kwargs, plot_separatrices, gridspec=gridspec, ax=ax)
kwargs_local['zorder'] = kwargs_local.get('zorder', sep_zorder)
out[0] += separatrices(
sys, pointdata, _check_kwargs=False, **kwargs_local)
sep_zorder = max(elem.get_zorder() for elem in out[0]) if out[0] \
else None
# Get rid of keyword arguments handled by separatrices
for kw in ['arrows', 'arrow_size', 'arrow_style', 'params']:
initial_kwargs.pop(kw, None)
equil_zorder = sep_zorder + 1 if sep_zorder else None
if plot_equilpoints:
kwargs_local = _create_kwargs(
kwargs, plot_equilpoints, gridspec=gridspec, ax=ax)
kwargs_local['zorder'] = kwargs_local.get('zorder', equil_zorder)
out[2] = equilpoints(
sys, pointdata, _check_kwargs=False, **kwargs_local)
# Get rid of keyword arguments handled by equilpoints
for kw in ['params']:
initial_kwargs.pop(kw, None)
# Make sure all keyword arguments were used
if initial_kwargs:
raise TypeError("unrecognized keywords: ", str(initial_kwargs))
if user_ax is None:
if title is None:
title = f"Phase portrait for {sys.name}"
_update_plot_title(title, use_existing=False, rcParams=rcParams)
ax.set_xlabel(sys.state_labels[0])
ax.set_ylabel(sys.state_labels[1])
plt.tight_layout()
return ControlPlot(out, ax, fig)
def vectorfield(
sys, pointdata, gridspec=None, zorder=None, ax=None,
suppress_warnings=False, _check_kwargs=True, **kwargs):
"""Plot a vector field in the phase plane.
This function plots a vector field for a two-dimensional state
space system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot the vector field in the given color.
ax : `matplotlib.axes.Axes`, optional
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : Quiver
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
zorder : float, optional
Set the zorder for the vectorfield. In not specified, it will be
automatically chosen by `matplotlib.axes.Axes.quiver`.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Determine the points on which to generate the vector field
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the plotting limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use
color = _get_color(kwargs, ax=ax)
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Generate phase plane (quiver) data
vfdata = np.zeros((points.shape[0], 4))
sys._update_params(params)
for i, x in enumerate(points):
vfdata[i, :2] = x
vfdata[i, 2:] = sys._rhs(0, x, np.zeros(sys.ninputs))
with plt.rc_context(rcParams):
out = ax.quiver(
vfdata[:, 0], vfdata[:, 1], vfdata[:, 2], vfdata[:, 3],
angles='xy', color=color, zorder=zorder)
return out
def streamplot(
sys, pointdata, gridspec=None, zorder=None, ax=None, vary_color=False,
vary_linewidth=False, cmap=None, norm=None, suppress_warnings=False,
_check_kwargs=True, **kwargs):
"""Plot streamlines in the phase plane.
This function plots the streamlines for a two-dimensional state
space system using the `matplotlib.axes.Axes.streamplot` function.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot.
gridspec : list, optional
Specifies the size of the grid in the x and y axes on which to
generate points.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot the vector field in the given color.
ax : `matplotlib.axes.Axes`, optional
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : StreamplotSet
Containter object with lines and arrows contained in the
streamplot. See `matplotlib.axes.Axes.streamplot` for details.
Other Parameters
----------------
cmap : str or Colormap, optional
Colormap to use for varying the color of the streamlines.
norm : `matplotlib.colors.Normalize`, optional
Normalization map to use for scaling the colormap and linewidths.
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.default['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
vary_color : bool, optional
If set to True, vary the color of the streamlines based on the
magnitude of the vector field.
vary_linewidth : bool, optional.
If set to True, vary the linewidth of the streamlines based on the
magnitude of the vector field.
zorder : float, optional
Set the zorder for the streamlines. In not specified, it will be
automatically chosen by `matplotlib.axes.Axes.streamplot`.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Determine the points on which to generate the streamplot field
points, gridspec = _make_points(pointdata, gridspec, 'meshgrid')
grid_arr_shape = gridspec[::-1]
xs = points[:, 0].reshape(grid_arr_shape)
ys = points[:, 1].reshape(grid_arr_shape)
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the plotting limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use
color = _get_color(kwargs, ax=ax)
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Generate phase plane (quiver) data
sys._update_params(params)
us_flat, vs_flat = np.transpose(
[sys._rhs(0, x, np.zeros(sys.ninputs)) for x in points])
us, vs = us_flat.reshape(grid_arr_shape), vs_flat.reshape(grid_arr_shape)
magnitudes = np.linalg.norm([us, vs], axis=0)
norm = norm or mpl.colors.Normalize()
normalized = norm(magnitudes)
cmap = plt.get_cmap(cmap)
with plt.rc_context(rcParams):
default_lw = plt.rcParams['lines.linewidth']
min_lw, max_lw = 0.25*default_lw, 2*default_lw
linewidths = normalized * (max_lw - min_lw) + min_lw \
if vary_linewidth else None
color = magnitudes if vary_color else color
out = ax.streamplot(
xs, ys, us, vs, color=color, linewidth=linewidths, cmap=cmap,
norm=norm, zorder=zorder)
return out
def streamlines(
sys, pointdata, timedata=1, gridspec=None, gridtype=None, dir=None,
zorder=None, ax=None, _check_kwargs=True, suppress_warnings=False,
**kwargs):
"""Plot stream lines in the phase plane.
This function plots stream lines for a two-dimensional state space
system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int, 1D array, or 2D array
Time to simulate each streamline. If a 1D array is given, then the
times at which to sample the simulation for all streamlines.
If a 2D array is given, then for each row the times at which
to sample the simulation for the corresponding streamline.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
dir : str, optional
Direction to draw streamlines: 'forward' to flow forward in time
from the reference points, 'reverse' to flow backward in time, or
'both' to flow both forward and backward. The amount of time to
simulate in each direction is given by the `timedata` argument.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : str
Plot the streamlines in the given color.
ax : `matplotlib.axes.Axes`, optional
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
arrows : int
Set the number of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrows']`.
arrow_size : float
Set the size of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_size']`.
arrow_style : matplotlib patch
Set the style of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_style']`.
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
zorder : float, optional
Set the zorder for the streamlines. In not specified, it will be
automatically chosen by `matplotlib.axes.Axes.plot`.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Parse the arrows keyword
arrow_pos, arrow_style = _parse_arrow_keywords(kwargs)
# Determine the points on which to generate the streamlines
points, gridspec = _make_points(pointdata, gridspec, gridtype=gridtype)
if dir is None:
dir = 'both' if gridtype == 'meshgrid' else 'forward'
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use
color = _get_color(kwargs, ax=ax)
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Create reverse time system, if needed
if dir != 'forward':
revsys = NonlinearIOSystem(
lambda t, x, u, params: -np.asarray(sys.updfcn(t, x, u, params)),
sys.outfcn, states=sys.nstates, inputs=sys.ninputs,
outputs=sys.noutputs, params=sys.params)
else:
revsys = None
# Generate phase plane (streamline) data
out = []
for i, X0 in enumerate(points):
# Create the trajectory for this point
timepts = _make_timepts(timedata, i)
traj = _create_trajectory(
sys, revsys, timepts, X0, params, dir,
gridtype=gridtype, gridspec=gridspec, xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
# Plot the trajectory (if there is one)
if traj.shape[1] > 1:
with plt.rc_context(rcParams):
out += ax.plot(traj[0], traj[1], color=color, zorder=zorder)
# Add arrows to the lines at specified intervals
_add_arrows_to_line2D(
ax, out[-1], arrow_pos, arrowstyle=arrow_style, dir=1)
return out
def equilpoints(
sys, pointdata, gridspec=None, color='k', zorder=None, ax=None,
_check_kwargs=True, **kwargs):
"""Plot equilibrium points in the phase plane.
This function plots the equilibrium points for a planar dynamical system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : str
Plot the equilibrium points in the given color.
ax : `matplotlib.axes.Axes`, optional
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
zorder : float, optional
Set the zorder for the equilibrium points. In not specified, it will
be automatically chosen by `matplotlib.axes.Axes.plot`.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Determine the points on which to generate the vector field
gridspec = [5, 5] if gridspec is None else gridspec
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Search for equilibrium points
equilpts = _find_equilpts(sys, points, params=params)
# Plot the equilibrium points
out = []
for xeq in equilpts:
with plt.rc_context(rcParams):
out += ax.plot(
xeq[0], xeq[1], marker='o', color=color, zorder=zorder)
return out
def separatrices(
sys, pointdata, timedata=None, gridspec=None, zorder=None, ax=None,
_check_kwargs=True, suppress_warnings=False, **kwargs):
"""Plot separatrices in the phase plane.
This function plots separatrices for a two-dimensional state space
system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int, 1D array, or 2D array
Time to simulate each streamline. If a 1D array is given, then the
times at which to sample the simulation for all streamlines.
If a 2D array is given, then for each row the times at which
to sample the simulation for the corresponding streamline.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot the separatrices in the given color. If a single color
specification is given, this is used for both stable and unstable
separatrices. If a tuple is given, the first element is used as
the color specification for stable separatrices and the second
element for unstable separatrices.
ax : `matplotlib.axes.Axes`, optional
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
zorder : float, optional
Set the zorder for the separatrices. In not specified, it will be
automatically chosen by `matplotlib.axes.Axes.plot`.
Notes
-----
The value of `config.defaults['separatrices_radius']` is used to set the
offset from the equilibrium point to the starting point of the separatix
traces, in the direction of the eigenvectors evaluated at that
equilibrium point.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Parse the arrows keyword
arrow_pos, arrow_style = _parse_arrow_keywords(kwargs)
# Determine the initial states to use in searching for equilibrium points
gridspec = [5, 5] if gridspec is None else gridspec
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Find the equilibrium points
equilpts = _find_equilpts(sys, points, params=params)
radius = config._get_param('phaseplot', 'separatrices_radius')
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use for stable, unstable subspaces
color = _get_color(kwargs)
match color:
case None:
stable_color = 'r'
unstable_color = 'b'
case (stable_color, unstable_color) | [stable_color, unstable_color]:
pass
case single_color:
stable_color = unstable_color = single_color
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Create a "reverse time" system to use for simulation
revsys = NonlinearIOSystem(
lambda t, x, u, params: -np.array(sys.updfcn(t, x, u, params)),
sys.outfcn, states=sys.nstates, inputs=sys.ninputs,
outputs=sys.noutputs, params=sys.params)
# Plot separatrices by flowing backwards in time along eigenspaces
out = []
for i, xeq in enumerate(equilpts):
# Figure out the linearization and eigenvectors
evals, evecs = np.linalg.eig(sys.linearize(xeq, 0, params=params).A)
# See if we have real eigenvalues (=> evecs are meaningful)
if evals[0].imag > 0:
continue
# Create default list of time points
if timedata is not None:
timepts = _make_timepts(timedata, i)
# Generate the traces
for j, dir in enumerate(evecs.T):
# Figure out time vector if not yet computed
if timedata is None:
timescale = math.log(maxlim / radius) / abs(evals[j].real)
timepts = np.linspace(0, timescale)
# Run the trajectory starting in eigenvector directions
dir = dir.real # use real components only
for eps in [-radius, radius]:
x0 = xeq + dir * eps
if evals[j].real < 0:
traj = _create_trajectory(
sys, revsys, timepts, x0, params, 'reverse',
gridtype='boxgrid', xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
color = stable_color
linestyle = '--'
elif evals[j].real > 0:
traj = _create_trajectory(
sys, revsys, timepts, x0, params, 'forward',
gridtype='boxgrid', xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
color = unstable_color
linestyle = '-'
# Plot the trajectory (if there is one)
if traj.shape[1] > 1:
with plt.rc_context(rcParams):
out += ax.plot(
traj[0], traj[1], color=color,
linestyle=linestyle, zorder=zorder)
# Add arrows to the lines at specified intervals
with plt.rc_context(rcParams):
_add_arrows_to_line2D(
ax, out[-1], arrow_pos, arrowstyle=arrow_style,
dir=1)
return out
#
# User accessible utility functions
#
# Utility function to generate boxgrid (in the form needed here)
def boxgrid(xvals, yvals):
"""Generate list of points along the edge of box.
points = boxgrid(xvals, yvals) generates a list of points that
corresponds to a grid given by the cross product of the x and y values.
Parameters
----------
xvals, yvals : 1D array_like
Array of points defining the points on the lower and left edges of
the box.
Returns
-------
grid : 2D array
Array with shape (p, 2) defining the points along the edges of the
box, where p is the number of points around the edge.
"""
return np.array(
[(x, yvals[0]) for x in xvals[:-1]] + # lower edge
[(xvals[-1], y) for y in yvals[:-1]] + # right edge
[(x, yvals[-1]) for x in xvals[:0:-1]] + # upper edge
[(xvals[0], y) for y in yvals[:0:-1]] # left edge
)
# Utility function to generate meshgrid (in the form needed here)
# TODO: add examples of using grid functions directly
def meshgrid(xvals, yvals):
"""Generate list of points forming a mesh.
points = meshgrid(xvals, yvals) generates a list of points that
corresponds to a grid given by the cross product of the x and y values.
Parameters
----------
xvals, yvals : 1D array_like
Array of points defining the points on the lower and left edges of
the box.
Returns
-------
grid : 2D array
Array of points with shape (n * m, 2) defining the mesh.
"""
xvals, yvals = np.meshgrid(xvals, yvals)
grid = np.zeros((xvals.shape[0] * xvals.shape[1], 2))
grid[:, 0] = xvals.reshape(-1)
grid[:, 1] = yvals.reshape(-1)
return grid
# Utility function to generate circular grid
def circlegrid(centers, radius, num):
"""Generate list of points around a circle.
points = circlegrid(centers, radius, num) generates a list of points
that form a circle around a list of centers.
Parameters
----------
centers : 2D array_like
Array of points with shape (p, 2) defining centers of the circles.
radius : float
Radius of the points to be generated around each center.