Diagrams

We describe here the modules used to group AUTO continuations into diagrams . Please read the User Guide to learn how to use them. TODO: make the user guide.

Bifurcation diagrams module

Bifurcation diagram class definition

This module defines the bifurcation diagram object used in auto-AUTO.

class auto2.diagrams.bifurcations.BifurcationDiagram(model_name=None, path_name=None)[source]

Bases: object

Base class for any continuation in auto-AUTO.

Parameters:

model_name (str) – The name of the model to load. Used to load the .f90 file of the provided model.
path_name (str, optional) – The directory path where files are read/saved. If None, defaults to current working directory.

initial_points

List of initial points used to start the bifurcation diagram. It has the following structure: Each point is an entry in the list and is specified with a dictionary with the following two keys:

‘parameters’: a dictionary of with the used continuation parameter and its value at which the initial_data (see below) is valid. I.e. its structure is {‘parameter_name’: parameter_value}.
‘initial_data’: a 1-D Numpy ndarray with the coordinates in phase space of the initial point, or an AUTOSolution object.

Type:: list(dict)

model_name

The name of the loaded model. Used to load the .f90 file of the provided model.

Type:: str

config_object

The ConfigParser object used during the continuations.

Type:: ConfigParser

fp_branches

Dictionary holding the continuations data of the fixed points of the bifurcation diagram, each labelled by their branch number (i.e. dictionary keys are the branch number). Each entry of the dictionary is itself a dictionary characterizing te branch and with the following entries:

‘parameters’: A list of the parameters values used to start the continuation(s).
‘continuation’: A Continuation object including the continuation(s) data.
‘continuation_kwargs’: A dictionary containing the AUTO parameters used to start the continuation(s).

Type:: dict

po_branches

Dictionary holding the continuations of the periodic orbits of the bifurcation diagram, each labelled by their branch number (i.e. dictionary keys are the branch number). Each entry of the dictionary is itself a dictionary characterizing te branch and with the following entries:

‘parameters’: A list of the parameters values used to start the continuation(s).
‘continuation’: A Continuation object including the continuation(s) data.
‘continuation_kwargs’: A dictionary containing the AUTO parameters used to start the continuation(s).

Type:: dict

fp_parent

Dictionary holding the branch number of the parents of a given fixed point branch (whose branch number is the dictionary key). None if there is no parent branch.

Type:: dict(int or None)

fp_branches_with_all_bp_computed

List of fixed point branch number for which all the branching points of the branch have been continued.

Type:: list(int)

fp_branches_with_all_hb_computed

List of fixed point branch number for which all the Hopf bifurcations of the branch have been continued.

Type:: list(int)

computed_bp_by_fp_branch

List of solution label number of branching points which have been continued for each fixed point branch (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second branching points on the forward continuation have been computed. Negative label numbers indicate branching points on the backward continuation.

Type:: dict(list(int))

computed_hb_by_fp_branch

List of solution label number of Hopf bifurcations which have been continued for each fixed point branch (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second Hopf points on the forward continuation have been computed. Negative label numbers indicate branching points on the backward continuation.

Type:: dict(list(int))

po_parent

Dictionary holding the branch number of the parents of a given periodic orbit branch (whose branch number is the dictionary key). None if there is no parent branch.

Type:: dict(int or None)

po_branches_with_all_bp_computed

List of periodic orbit branch number for which all the branching points of the branch have been continued.

Type:: list(int)

po_branches_with_all_pd_computed

List of periodic orbit branch number for which all the period doubling bifurcations of the branch have been continued.

Type:: list(int)

computed_bp_by_po_branch

List of solution label number of branching points which have been continued for each periodic orbit (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second branching points on the forward continuation have been computed. Negative label numbers indicate branching points on the backward continuation.

Type:: dict(list(int))

valid_bp_by_po_branch: List of solution label number periodic orbit branching points which have resulted in a valid continuation for each periodic orbit (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second periodic orbit branching points on the forward continuation were valid. Negative label numbers indicate branching points on the backward continuation. A valid continuation is a continuation for which no subset is identified with another branch or another part of itself (loops).

computed_pd_by_po_branch

List of solution label number of period doubling bifurcations which have been continued for each periodic orbit (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second period doubling points on the forward continuation have been computed. Negative label numbers indicate branching points on the backward continuation.

Type:: dict(list(int))

valid_pd_by_po_branch

List of solution label number of period doubling bifurcations which have resulted in a valid continuation for each periodic orbit (indexed by their branch number forming the dictionary keys). E.g. for a given branch, [1, 2] indicates that the branches emanating from the first and second period doubling points on the forward continuation were valid. Negative label numbers indicate branching points on the backward continuation. A valid continuation is a continuation for which no subset is identified with another branch or another part of itself (loops).

Type:: dict(list(int))

fp_computed

Whether all the fixed points continuations possibilities of the bifurcation diagram have been computed.

Type:: bool

po_computed

Whether all the periodic orbits continuations possibilities of the bifurcation diagram have been computed.

Type:: bool

add_periodic_orbit(initial_data, extra_comparison_parameters=None, comparison_tol=0.02, max_number_bp_detected=None, only_forward=False, **continuation_kwargs)[source]

Continue and then add manually a given periodic orbit to the periodic orbit bifurcation diagram.

Parameters:

initial_data (AUTOSolution or str) – Initial data used to start the continuation(s). Should be an AUTOSolution or a string indicating the path to a file containing the data of the periodic orbit to start from. See the dat parameter in the AUTO parameters documentation for more detail about how this data file must be organized (you can also find this documented below).
extra_comparison_parameters (list(str or int) or None, optional) – List of extra parameters labels or numbers to use in the various comparison to determine the validity of the continuations. If None, only the defaults parameters will be considered, i.e. the continuation parameter and the L2 norm. Otherwise, the specified parameters are added to the two default ones. Default to None.
comparison_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the parameters comparison done to check the continuations validity. If a single float is provided, assume the same tolerance for each parameter. If a list or a 1-D array is passed, it must have the dimension 2+len(extra_comparison_parameters), each value in the array corresponding to a parameter. Default to 0.02.
max_number_bp_detected (int or None, optional) – Number of branching points detected during the continuations of periodic orbits, before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.
only_forward (bool, optional) – If True, compute only the forward continuation (positive DS parameter). If False, compute in both backward and forward direction. Default to False.
continuation_kwargs (dict) – Parameters to pass to AUTO for each continuation. See the make_continuation() method of the PeriodicOrbitContinuation class for more details about the available AUTO parameters.

compute_fixed_points_diagram(initial_points=None, extra_comparison_parameters=None, comparison_tol=0.02, **continuation_kwargs)[source]

Method which starts the computation of a fixed points bifurcation diagrams using a set of provided initial points.

Parameters:

initial_points (list(dict) or None, optional) –
List of initial fixed points to consider for computing the diagram. Each entry in the list is a dictionary with the following two keys:
- ’parameters’: a dictionary of with the used continuation parameter and its value at which the initial_data (see below) is valid. I.e. its structure is {‘parameter_name’: parameter_value}. Can be overloaded by the values found in the continuation_kwargs.
- ’initial_data’: a 1-D Numpy ndarray with the coordinates in phase space of the initial point, or an AUTOSolution object.
If None, recompute the fixed points already present in the initial_points class attribute. Default to None.
extra_comparison_parameters (list(str or int) or None, optional) – List of extra parameters labels or numbers to use in the various comparison to determine the validity of the continuations. If None, only the defaults parameters will be considered, i.e. the continuation parameter and the L2 norm. Otherwise, the specified parameters are added to the two default ones. Default to None.
comparison_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the parameters comparison done to check the continuations validity. If a single float is provided, assume the same tolerance for each parameter. If a list or a 1-D array is passed, it must have the dimension 2+len(extra_comparison_parameters), each value in the array corresponding to a parameter. Default to 0.02.
continuation_kwargs (dict) – Parameters to pass to AUTO for each continuation. See the make_continuation() method of the FixedPointContinuation class for more details about the available AUTO parameters.

compute_periodic_orbits_diagram(end_level=10, extra_comparison_parameters=None, comparison_tol=0.02, remove_dubious_bp=True, max_number_bp=None, max_number_bp_detected=None, backward_bp_continuation=False, **continuation_kwargs)[source]

Method which starts the computation of the periodic orbits of a bifurcation diagram where the fixed points have already been continued. It will start by continuing the Hopf bifurcations, then the resulting branching and period doubling points. Each computation is done level by level in the the developing tree of the bifurcation diagram. For instance, fixed point continuation is the level 0, and the continuations of the Hopf bifurcations form the level 1. Computation stops either if all the bifurcation points have been computed/continued, or if a specified level of the tree has been reached.

Parameters:

end_level (int, optional) – Level of the periodic orbits bifurcation diagram tree where the computation must stop. Defaults to 10.
extra_comparison_parameters (list(str or int) or None, optional) – List of extra parameters labels or numbers to use in the various comparison to determine the validity of the continuations. If None, only the defaults parameters will be considered, i.e. the continuation parameter and the L2 norm. Otherwise, the specified parameters are added to the two default ones. Default to None.
comparison_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the parameters comparison done to check the continuations validity. If a single float is provided, assume the same tolerance for each parameter. If a list or a 1-D array is passed, it must have the dimension 2+len(extra_comparison_parameters), each value in the array corresponding to a parameter. Default to 0.02.
remove_dubious_bp (bool, optional) – Do not compute continuation of branching points for which the stability information include HUGE numbers, indicating a possible problem in the parent continuation.
max_number_bp (int or None, optional) – Only continue the first max_number_bp branching points of the parent branches. If None, continue all the branching points of the parent branches. Defaults to None.
max_number_bp_detected (int or None, optional) – Number of branching points detected during the continuations of periodic orbits, before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.
backward_bp_continuation (bool, optional) – Compute also the backward continuation at the branching points. Defaults to False.
continuation_kwargs (dict) – Parameters to pass to AUTO for each continuation. See the make_continuation() method of the PeriodicOrbitContinuation class for more details about the available AUTO parameters.

property fixed_points_solutions_list

list of all the fixed points solutions of the bifurcation diagram.

Type:: list(AUTOSolution)

property fixed_points_variables_list

Returns the list of variable names used in the fixed points continuations.

Type:: list(str)

property full_solutions_list

list of all the solutions of the bifurcation diagram.

Type:: list(AUTOSolution)

get_continuation(idx)[source]

Get the continuation dictionary of a give branch, indexed by his branch number.

Parameters:: idx (int) – The branch number of the sought branch continuation dictionary.

load(filename, load_initial_data=True, **kwargs)[source]

Load the AUTO files and the branch parameters and metadata corresponding to a whole bifurcation diagram.

Parameters:

filename (str) – The filename from which to load Pickle file with the parameters and metadata of the bifurcation diagram.
load_initial_data (bool, optional) – Load nor not the initial data into the BifurcationDiagram object. Default to True.
kwargs (dict, optional) – Keyword arguments passed to Pickle.

property number_of_fp_branches

Number of fixed points continuation branches in the bifurcation diagram.

Type:: int

property number_of_po_branches

Number of periodic orbits continuation branches in the bifurcation diagram.

Type:: int

property path_name

The path where the AUTO and AUTO² files must be or are stored.

Type:: str

property periodic_orbits_solutions_list

list of all the periodic orbits solutions of the bifurcation diagram.

Type:: list(AUTOSolution)

property periodic_orbits_variables_list

Returns the list of variable names used in the periodic orbits continuations.

Type:: list(str)

property phase_space_variables

Return the variables of the phase space of the configured dynamical system.

Type:: list(str)

plot_diagram_and_solutions(solutions_parameter_value, diagram_variables=(1,), solutions_variables=(0, 1), axes=None, figsize=(10, 16), solutions_tol=0.01, fixed_points_diagram_kwargs=None, periodic_orbits_diagram_kwargs=None, solutions_kwargs=None)[source]

Plots a bifurcation diagram including all fixed point and periodic orbit branches, along with a projection of the solutions at a given parameter value.

Parameters:

solutions_parameter_value (float) – The parameter value for which the solutions are plot. The parameter that is used is governed by the continuation parameter that was used.
diagram_variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by fixed_points_variables_list() and periodic_orbits_variables_list(). The first value in the tuple determines the variable plot on the y-axis. Defaults to (1, ), which plots the L2 Norm.
solutions_variables (tuple(int or str, int or str), optional) – The index or label of the variables shown in the solution plot. If labels are used, it should be labels of the phase space variables (if they have been defined in the AUTO configuration file) given by phase_space_variables(). The first value in the tuple determines the variable plot on the x-axis, the second determines the y-axis. Defaults to the first two variables: (0, 1).
axes (ndarray(matplotlib.axes.Axes) or None, optional) – An array of two Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 16).
solutions_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance used when comparing solutions to display with the selected solutions_parameter_value argument. Default to 0.01.
fixed_points_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the fixed point bifurcation diagram plotting function plot_fixed_points_diagram().
periodic_orbits_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the periodic orbit bifurcation diagram plotting function plot_periodic_orbits_diagram().
solutions_kwargs (dict or None, optional) – Keyword arguments to be passed to the solution plotting function plot_solutions() of the continuation objects.

The parameters below can be included in solutions_kwargs for controlling the solution plots. See auto2.continuations.base.Continuation.plot_solutions() for more details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
marker (str, optional) – The marker style used for plotting fixed points. If None, set to default marker.
linestyle (str, optional) – The line style used for plotting the periodic orbit solutions. If None, set to default marker.
linewidth (float, optional) – The width of the lines showing the periodic solutions. If None, set to the default.
color_solutions (bool, optional) – Whether to color each solution differently. If True, and parameter argument is provided and valid, then solutions are colored based on the parameter value of a given solution. Use the cmap provided in the plot_kwargs argument. If no cmap is provided there, use Blues cmap by default. If False, all solutions are colored identically, controlled using plot_kwargs. Default is False.
plot_kwargs (dict or None, optional) – Additional keyword arguments for the plotting function. Default is None.
variables_name (list(str) or None, optional) – The strings used to plot the axis labels. If None defaults to the AUTO labels. Defaults to None.

Returns:

An array of two Matplotlib axis objects, showing the bifurcation diagram and the solution plot.

Return type:

ndarray(matplotlib.axes.Axes)

plot_diagram_and_solutions_in_3D(solutions_parameter_value, diagram_variables=(1,), solutions_variables=(0, 1, 2), axes=None, figsize=(10, 16), solutions_tol=0.01, fixed_points_diagram_kwargs=None, periodic_orbits_diagram_kwargs=None, solutions_kwargs=None)[source]

Plots a bifurcation diagram including all fixed point and periodic orbit branches using a 2-dimensional projection,: along with a projection of the solutions at a given parameter value that uses a 3-dimensional projection.

Parameters:

solutions_parameter_value (float) – The parameter value for which the solutions are plot. The parameter that is used is governed by the continuation parameter that was used.
diagram_variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by fixed_points_variables_list() and periodic_orbits_variables_list(). The first value in the tuple determines the variable plot on the y-axis. Defaults to (1, ), which plots the L2 Norm.
solutions_variables (tuple(int or str, int or str, int or str), optional) – The index or label of the variables shown in the solution plot. If labels are used, it should be labels of the phase space variables (if they have been defined in the AUTO configuration file) given by phase_space_variables(). The first value in the tuple determines the variable plot on the x-axis, the second determines the y-axis, and the last value determines the z-axis Defaults to the first two variables: (0, 1, 2).
axes (ndarray(matplotlib.axes.Axes) or None, optional) – An array of two Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 16).
solutions_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance used when comparing solutions to display with the selected solutions_parameter_value argument. Default to 0.01.
fixed_points_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the fixed point bifurcation diagram plotting function plot_fixed_points_diagram().
periodic_orbits_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the periodic orbit bifurcation diagram plotting function plot_periodic_orbits_diagram().
solutions_kwargs (dict or None, optional) – Keyword arguments to be passed to the solution plotting function plot_solutions_3D() of the continuation objects.

The parameters below can be included in solutions_kwargs for controlling the solution plots. See auto2.continuations.base.Continuation.plot_solutions_3D() for more details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
marker (str, optional) – The marker style used for plotting fixed points. If None, set to default marker.
linestyle (str, optional) – The line style used for plotting the periodic orbit solutions. If None, set to default marker.
linewidth (float, optional) – The width of the lines showing the periodic solutions. If None, set to the default.
plot_kwargs (dict or None, optional) – Additional keyword arguments for the plotting function. Default is None.
variables_name (list(str) or None, optional) – The strings used to plot the axis labels. If None defaults to the AUTO labels. Defaults to None.

Returns:

An array of two Matplotlib axis objects, showing the bifurcation diagram and the 3-dimensional solution plot.

Return type:

numpy.ndarray(matplotlib.axes.Axes)

plot_diagram_in_3D_and_solutions_in_3D(solutions_parameter_value, diagram_variables=(1, 2), solutions_variables=(0, 1, 2), axes=None, figsize=(10, 16), solutions_tol=0.01, fixed_points_diagram_kwargs=None, periodic_orbits_diagram_kwargs=None, solutions_kwargs=None)[source]

Plots a 3-dimensional projection of the bifurcation diagram including all fixed point and periodic orbit branches, along with a projection of the solutions at a given parameter value, also using a 3-dimensional projection.

Parameters:

solutions_parameter_value (float) – The parameter value for which the solutions are plot. The parameter that is used is governed by the continuation parameter that was used.
diagram_variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by fixed_points_variables_list() and periodic_orbits_variables_list(). The first value in the tuple determines the variable plot on the x-axis, the second the y-axis. Defaults to (1, 2), which plots the L2 Norm and the first variable of the phase space.
solutions_variables (tuple(int or str, int or str, int or str), optional) – The index or label of the variables shown in the solution plot. If labels are used, it should be labels of the phase space variables (if they have been defined in the AUTO configuration file) given by phase_space_variables(). The first value in the tuple determines the variable plot on the x-axis, the second determines the y-axis, and the last value determines the z-axis Defaults to the first two variables: (0, 1, 2).
axes (ndarray(matplotlib.axes.Axes) or None, optional) – An array of two Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 16).
solutions_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance used when comparing solutions to display with the selected solutions_parameter_value argument. Default to 0.01.
fixed_points_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the fixed point bifurcation diagram plotting function plot_fixed_points_diagram_3D().
periodic_orbits_diagram_kwargs (dict or None, optional) – Keyword arguments to be passed to the periodic orbit bifurcation diagram plotting function plot_periodic_orbits_diagram_3D().
solutions_kwargs (dict or None, optional) – Keyword arguments to be passed to the solution plotting function plot_solutions_3D() of the continuation objects.

The parameters below can be included in solutions_kwargs for controlling the solution plots. See auto2.continuations.base.Continuation.plot_solutions_3D() for more details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
marker (str, optional) – The marker style used for plotting fixed points. If None, set to default marker.
linestyle (str, optional) – The line style used for plotting the periodic orbit solutions. If None, set to default marker.
linewidth (float, optional) – The width of the lines showing the periodic solutions. If None, set to the default.
plot_kwargs (dict or None, optional) – Additional keyword arguments for the plotting function. Default is None.
variables_name (list(str) or None, optional) – The strings used to plot the axis labels. If None defaults to the AUTO labels. Defaults to None.

Returns:

An array of two Matplotlib axis objects, showing the 3-dimensional bifurcation diagram and the solution plot.

Return type:

ndarray(matplotlib.axes.Axes)

plot_fixed_points_diagram(variables=(0, 1), ax=None, figsize=(10, 8), cmap=None, return_used_colors=False, legend=True, **kwargs)[source]

Plots the bifurcation diagram of all the fixed point branches.

Parameters:

variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by fixed_points_variables_list(). The first value in the tuple determines the variable plot on the x-axis and the second the y-axis. Defaults to (0, 1).
ax (matplotlib.axes.Axes or None, optional) – A Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 8).
cmap (matplotlib.colors.Colormap or str or None, optional) – A colormap that controls the color of the branches. The branches are colored from the colormap based on the value of the branch number. If None, defaults to the Matplotlib Tableau colors list.
return_used_colors (bool, optional) – If True, the colors used in the diagram are returned. Defaults to False.
legend (bool, optional) – If True, a legend of the branches is included in the plot. Defaults to False.
kwargs (dict, optional) – Keyword arguments to be passed to the plot_branch_parts() method of the continuation objects. See below for further details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
plot_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.
marker_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the styles of the bifurcation point labels.
excluded_labels (str or list(str), optional) – A list of 2-characters strings, controlling the bifurcation point type that are not plotted. For example [‘BP’] will result in branching points not being plotted. Can be set to the string ‘all’, which results in no bifurcation being plotted at all. Default to [‘UZ’,’EP’].
plot_sol_points (bool, optional) – If True, a point is added to the points where a solution is stored. Defaults to False.
variables_name (list(str) or None, optional) – The strings used to define the axis labels. If None defaults to the AUTO labels.

Returns:

A Matplotlib axis object showing the fixed-point bifurcation diagram.

Return type:

matplotlib.axes.Axes

plot_fixed_points_diagram_3D(variables=(0, 1, 2), ax=None, figsize=(10, 8), cmap=None, return_used_colors=False, legend=True, **kwargs)[source]

Plots the bifurcation diagram of all the fixed point branches using a 3-dimensional projection.

Parameters:

variables (tuple(int or str, int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by fixed_points_variables_list(). The first value in the tuple determines the variable plot on the x-axis, the second the y-axis, and the last value determines the z-axis. Defaults to (0, 1, 2).
ax (matplotlib.axes.Axes or None, optional) – A Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 8).
cmap (matplotlib.colors.Colormap or str or None, optional) – A colormap that controls the color of the branches. The branches are colored from the colormap based on the value of the branch number. If None, defaults to the Matplotlib Tableau colors list.
return_used_colors (bool, optional) – If True, the colors used in the diagram are returned. Defaults to False.
legend (bool, optional) – If True, a legend of the branches is included in the plot. Defaults to False.
kwargs (dict, optional) – Keyword arguments to be passed to the plot_branch_parts() method of the continuation objects. See below for further details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
plot_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.
marker_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the styles of the bifurcation point labels.
excluded_labels (str or list(str), optional) – A list of 2-characters strings, controlling the bifurcation point type that are not plotted. For example [‘BP’] will result in branching points not being plotted. Can be set to the string ‘all’, which results in no bifurcation being plotted at all. Default to [‘UZ’,’EP’].
plot_sol_points (bool, optional) – If True, a point is added to the points where a solution is stored. Defaults to False.
variables_name (list(str) or None, optional) – The strings used to define the axis labels. If None defaults to the AUTO labels.

Returns:

A Matplotlib axis object showing the fixed-point bifurcation diagram using a 3-dimensional projection.

Return type:

matplotlib.axes.Axes

plot_periodic_orbits_diagram(variables=(0, 1), ax=None, figsize=(10, 8), cmap=None, return_used_colors=False, legend=True, **kwargs)[source]

Plots the bifurcation diagram of all the periodic orbit branches.

Parameters:

variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by periodic_orbits_variables_list(). The first value in the tuple determines the variable plot on the x-axis and the second the y-axis. Defaults to (0, 1).
ax (matplotlib.axes.Axes or None, optional) – A Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 8).
cmap (matplotlib.colors.Colormap or str or None, optional) – A colormap that controls the color of the branches. The branches are colored from the colormap based on the value of the branch number. If None, defaults to the Matplotlib Tableau colors list.
return_used_colors (bool, optional) – If True, the colors used in the diagram are returned. Defaults to False.
legend (bool, optional) – If True, a legend of the branches is included in the plot. Defaults to False.
kwargs (dict, optional) – Keyword arguments to be passed to the plot_branch_parts() method of the continuation objects. See below for further details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
plot_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.
marker_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the styles of the bifurcation point labels.
excluded_labels (str or list(str), optional) – A list of 2-characters strings, controlling the bifurcation point type that are not plotted. For example [‘BP’] will result in branching points not being plotted. Can be set to the string ‘all’, which results in no bifurcation being plotted at all. Default to [‘UZ’,’EP’].
plot_sol_points (bool, optional) – If True, a point is added to the points where a solution is stored. Defaults to False.
variables_name (list(str) or None, optional) – The strings used to define the axis labels. If None defaults to the AUTO labels.

Returns:

A Matplotlib axis object showing the periodic-orbit bifurcation diagram.

Return type:

matplotlib.axes.Axes

plot_periodic_orbits_diagram_3D(variables=(0, 1, 2), ax=None, figsize=(10, 8), cmap=None, return_used_colors=False, legend=True, **kwargs)[source]

Plots the bifurcation diagram of all the periodic orbit branches using a 3-dimensional projection.

Parameters:

variables (tuple(int or str, int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by periodic_orbits_variables_list(). The first value in the tuple determines the variable plot on the x-axis, the second the y-axis, and the last value determines the z-axis. Defaults to (0, 1, 2).
ax (matplotlib.axes.Axes or None, optional) – A Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 8).
cmap (matplotlib.colors.Colormap or str or None, optional) – A colormap that controls the color of the branches. The branches are colored from the colormap based on the value of the branch number. If None, defaults to the Matplotlib Tableau colors list.
return_used_colors (bool, optional) – If True, the colors used in the diagram are returned. Defaults to False.
legend (bool, optional) – If True, a legend of the branches is included in the plot. Defaults to False.
kwargs (dict, optional) – Keyword arguments to be passed to the plot_branch_parts() method of the continuation objects. See below for further details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
plot_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.
marker_kwargs (dict or None, optional) – Keyword arguments to pass to the plotting function, controlling the styles of the bifurcation point labels.
excluded_labels (str or list(str), optional) – A list of 2-characters strings, controlling the bifurcation point type that are not plotted. For example [‘BP’] will result in branching points not being plotted. Can be set to the string ‘all’, which results in no bifurcation being plotted at all. Default to [‘UZ’,’EP’].
plot_sol_points (bool, optional) – If True, a point is added to the points where a solution is stored. Defaults to False.
variables_name (list(str) or None, optional) – The strings used to define the axis labels. If None defaults to the AUTO labels.

Returns:

A Matplotlib axis object showing the periodic-orbit bifurcation diagram using a 3-dimensional projection.

Return type:

matplotlib.axes.Axes

plot_single_po_branch_and_solutions(branch_number, parameter=None, diagram_variables=(1,), solutions_variables=(0, 1), axes=None, figsize=(10, 16), solutions_tol=0.01, cmap=None, branch_diagram_kwargs=None, solution_diagram_kwargs=None)[source]

Plots the bifurcation diagram of a single branch with the parameter values of stored solutions plotted on this branch, along with a plot of the projection of all solutions.

Parameters:

branch_number (int) – The number of the branch to plot.
parameter (str, optional) – The parameter along which to plot the diagram and solutions.
diagram_variables (tuple(int or str, int or str), optional) – The index or label of the variables along which the bifurcation diagram is plotted. If labels are used, it should be labels of the available monitored variables during the continuations (if their labels have been defined in the AUTO configuration file) given by available_variables(). The first value in the tuple determines the variable plot on the x-axis, and the second the y-axis Defaults to (1, ).
solutions_variables (tuple(int or str, int or str), optional) – The index or label of the variables shown in the solution plot. If labels are used, it should be labels of the phase space variables (if they have been defined in the AUTO configuration file) given by phase_space_variables(). The first value in the tuple determines the variable plot on the x-axis, the second determines the y-axis. Defaults to the first two variables: (0, 1).
axes (numpy.ndarray(matplotlib.axes.Axes) or None, optional) – An array of two Matplotlib axis object to plot on. If None, a new figure is created.
figsize (tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 16).
solutions_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance used when comparing solutions to display with the selected solutions_parameter_value. Default to 0.01.
cmap (matplotlib.colors.Colormap or str or None, optional) – A colormap that controls the color of the branches. The branches are colored from the colormap based on the value of the branch number. If None, defaults to Reds colormaps.
branch_diagram_kwargs (dict or None, optional) – Additional keyword arguments for the bifurcation diagram plotting function. Default is None.
solution_diagram_kwargs (dict or None, optional) – Additional keyword arguments for the solution plotting function. Default is None.

The parameters below can be included in solutions_kwargs for controlling the solution plots. See auto2.continuations.base.Continuation.plot_solutions() for more details.

Keyword Arguments:

markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.
marker (str, optional) – The marker style used for plotting fixed points. If None, set to default marker.
linestyle (str, optional) – The line style used for plotting the periodic orbit solutions. If None, set to default marker.
linewidth (float, optional) – The width of the lines showing the periodic solutions. If None, set to the default.
color_solutions (bool, optional) – Whether to color each solution differently. If True, and parameter argument is provided and valid, then solutions are colored based on the parameter value of a given solution. Use the cmap provided in the plot_kwargs argument. If no cmap is provided there, use Blues cmap by default. If False, all solutions are colored identically, controlled using plot_kwargs. Default is False.
plot_kwargs (dict or None, optional) – Additional keyword arguments for the plotting function. Default is None.
variables_name (list(str) or None, optional) – The strings used to plot the axis labels. If None defaults to the AUTO labels. Defaults to None.

Returns:

An array of two Matplotlib axis objects, showing the bifurcation diagram and the solution plot.

Return type:

numpy.ndarray(matplotlib.axes.Axes)

restart_periodic_orbits_diagram(end_level=10, extra_comparison_parameters=None, comparison_tol=0.02, remove_dubious_bp=True, max_number_bp=None, max_number_bp_detected=None, backward_bp_continuation=False, restart=True, **continuation_kwargs)[source]

Method which restarts the computation of the periodic orbits of a bifurcation diagram stopped previously at a certain level of the developing tree of the bifurcation diagram. Each computation is done level by level in the the developing tree of the bifurcation diagram. For instance, fixed point continuation is the level 0, and the continuations of the Hopf bifurcations form the level 1. Computation stops either if all the bifurcation points have been computed/continued, or if a specified level of the tree has been reached.

Parameters:

end_level (int, optional) – Level of the periodic orbits bifurcation diagram tree where the computation must stop. Defaults to 10.
extra_comparison_parameters (list(str or int) or None, optional) – List of extra parameters labels or numbers to use in the various comparison to determine the validity of the continuations. If None, only the defaults parameters will be considered, i.e. the continuation parameter and the L2 norm. Otherwise, the specified parameters are added to the two default ones. Default to None.
comparison_tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the parameters comparison done to check the continuations validity. If a single float is provided, assume the same tolerance for each parameter. If a list or a 1-D array is passed, it must have the dimension 2+len(extra_comparison_parameters), each value in the array corresponding to a parameter. Default to 0.02.
remove_dubious_bp (bool, optional) – Do not compute continuation of branching points for which the stability information include HUGE numbers, indicating a possible problem in the parent continuation.
max_number_bp (int or None, optional) – Only continue the first max_number_bp branching points of the parent branches. If None, continue all the branching points of the parent branches. Defaults to None.
max_number_bp_detected (int or None, optional) – Number of branching points detected during the continuations of periodic orbits, before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.
backward_bp_continuation (bool, optional) – Compute also the backward continuation at the branching points. Defaults to False.
restart (bool, optional) – Whether to restart or simply start the computation of the periodic orbits bifurcation diagram. Defaults to True which means restart.
continuation_kwargs (dict) – Parameters to pass to AUTO for each continuation. See the make_continuation() method of the PeriodicOrbitContinuation class for more details about the available AUTO parameters.

save(filename=None, **kwargs)[source]

Save the AUTO files and the branch parameters and metadata for the whole bifurcation diagram. The continuation branches parameters and metadata will be stored in a Pickle file according to given naming convention. The data and metadata of the bifurcation diagram will be stored in a separate Pickle file with a provided file name.

Parameters:

filename (str or None, optional) – The filename used to store parameters and metadata of the bifurcation diagram to disk in Pickle format. If None, will assign a default generic name. Default is None.
kwargs (dict, optional) – Keyword arguments passed to Pickle.

set_path_name(path_name)[source]

Set the path where the AUTO and AUTO² files must be or are stored.

Parameters:: path_name (str) – The path.