Continuations

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

Base class module

Continuation base class definition

This module implements the definition used by any subsequent continuation subclass in auto-AUTO.

class auto2.continuations.base.Continuation(model_name, config_object, path_name=None)[source]

Bases: ABC

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.

  • config_object (ConfigParser) – A loaded ConfigParser object.

  • path_name (str, optional) – The directory path where files are read/saved. If None, defaults to current working directory.

model_name

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

Type:

str

config_object

A loaded ConfigParser object.

Type:

ConfigParser

continuation

Dictionary holding the forward and backward continuation data.

Type:

dict

branch_number

The AUTO branch number attributed to the continuation(s). This number can be set manually by passing the IBR AUTO parameter when starting the continuations (see the make_continuation() documentation for more details).

Type:

int

initial_data

The initial data used to start the continuation(s).

Type:

ndarray or AUTOSolution object

auto_filename_suffix

Suffix for the AUTO files used to save the continuation(s) data and parameters on disk.

Type:

str

auto_load(auto_suffix)[source]

Load the AUTO files for both the forward and backward continuations if they exist.

Parameters:

auto_suffix (str) – Suffix to use for the AUTO files used to load the continuation data from disk.

auto_save(auto_suffix)[source]

Save the AUTO files for both the forward and backward continuations if they exist.

Parameters:

auto_suffix (str) – Suffix to use for the AUTO files used to save the continuation data on disk.

property available_variables

Return the available variables in the continuation data.

Type:

list(str)

branch_possibly_cross(other, parameters, solutions_types=('HB', 'BP', 'UZ', 'PD', 'TR', 'LP'), tol=0.02, return_parameters=False, return_solutions=False, forward=None)[source]

Check if the continuations of two Continuation objects are possibly crossing at a given bifurcation point.

Warning

As indicated by the possibly, this test is weak and can be wrong.

Parameters:

  • other (Continuation) – The other continuation object to check the solutions.

  • parameters (list(str)) – The parameters to be considered to check if the solutions are the same.

  • solutions_types (list(str)) – The types of solution to consider in the search. Default to [‘HB’, ‘BP’, ‘UZ’, ‘PD’, ‘TR’, ‘LP’].

  • tol (float or list(float) or ndarray(float)) – The numerical tolerance of the comparison to determine if two solutions are equal. 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 of the number of parameters, each value in the array corresponding to a parameter.

  • return_parameters (bool) – If True, return an additional array with the values of the parameters for which the solutions of the two Continuation objects match.

  • return_solutions (bool) – If True, return an additional list with the AUTO solution objects.

  • forward (bool or None, optional) – If True, search only in the forward continuation. If False, search only in the backward continuation. If None, search in both backward and forward direction. Default to None.

Returns:

  • crossing (bool) – True if the solutions are all contained in the other continuations according to the specified parameters.

  • crossing_values (~numpy.ndarray) – If return_parameters is True, an array with the values of the parameters for which the solutions of the two continuations match.

  • solutions_list (list(AUTOSolution object)) – If return_solutions is True, a list of the solutions which are included in the other Continuation object.

check_for_repetitions(parameters, tol=0.02, return_parameters=False, return_non_repeating_solutions=False, return_repeating_solutions=False, forward=None)[source]

Check in solutions in the Continuation object are repeating, possibly indicating that the continuation is looping on itself.

Parameters:

  • parameters (list(str)) – The parameters to be considered to check if the solutions are the same.

  • tol (float or list(float) or ndarray(float)) – The numerical tolerance of the comparison to determine if two solutions are equal. 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 of the number of parameters, each value in the array corresponding to a parameter.

  • return_parameters (bool) – If True, return an additional array with the values of the parameters for which the solutions are repeating.

  • return_repeating_solutions (bool) – If True, return an additional list with the AUTO solution objects which repeat.

  • return_non_repeating_solutions (bool) – If True, return an additional list with the AUTO solution objects which do nott repeat.

  • forward (bool or None, optional) – If True, search only in the forward continuation. If False, search only in the backward continuation. If None, search in both backward and forward direction. Default to None.

Returns:

  • repeating (dict(list(bool))) – A dictionary with a list of boolean for each direction, indicating for each solution if it is repeating or not.

  • repeating_values (dict(~numpy.ndarray)) – If return_parameters is True, a dictionary with for each direction an array with the values of the parameters for the solutions are repeating.

  • non_repeating_solutions_list (dict(list(AUTOSolution object))) – If return_non_repeating_solutions is True, a dictionary with for each direction a list of the solutions which are included which do not repeat.

  • repeating_solutions_list (dict(list(AUTOSolution object))) – If return_repeating_solutions is True, a dictionary with for each direction a list of the solutions which are included which do repeat.

property continuation_parameters

The continuation parameters of both forward and backward continuations.

Type:

dict(list(str))

diagnostics()[source]

str: Return the AUTO diagnostics of the continuations as a string.

find_solution_index(label)[source]

Find the index of a given solution label.

Parameters:

label (str) – A string with the label of the point for which to return the solution object. Backward continuation label must be preceded by a ‘-’ character. E.g. ‘HB2’ identifies the second Hopf bifurcation point of the forward continuation, while ‘-UZ3’ identifies the third user-defined label of the backward branch.

Returns:

The AUTO index of the provided label. Positive integers identify points on the forward continuation, while negative integers identify points on the backward continuation. The zero index identifies the initial point of the continuations.

Return type:

int

property full_solutions_list

The full list of solutions of the branch ordered from the last solution of the backward continuation to the last solution of the forward continuation.

Type:

list(AUTOSolution object)

property full_solutions_list_by_label

The full list of solutions sorted by labels of both forward and backward continuations.

Type:

list(AUTOSolution object)

get_filtered_solutions_list(labels=None, indices=None, parameters=None, values=None, forward=None, tol=0.01)[source]

Filter the full solutions list of the branch with different selection rules: * Based on solution labels * Based on solution indices * Based matching solution parameters values with a specified list of values

Selection rules are selected by specifying a given keyword. If no selection rule is provided, return the full list of solution in the selected direction(s).

Parameters:

  • labels (str or list(str), optional) – Label or list of labels to look for the solutions. Labels are two-characters string specifying the solutions types to return. For example ‘BP’ will return all the branching points of the continuations. This activates the selection rule based on labels and disable the others.

  • indices (int or list(int), optional) – Index or list of indices to look for the solutions. AUTO index of solutions can be inquired for example by calling the print_summary() method of a given Continuation object. This activates the selection rule based on indices and disable the others.

  • parameters (str or list(str) or ndarray(str), optional) – Parameter or list of parameters for which to match the solutions values to one provided by the values argument.

  • values (float or list(float) or ndarray(float), optional) – List of parameters values to match to the solutions parameters values. Can be a float if only one parameter is specified in the parameters arguments, otherwise a list or a ndarray array must be provided. If a list is provided, assuming n parameters were specified in the parameters argument and`m` values are sought, it should be a nested list with the following structure: [[param1_value1, param1_value2, …, param1_valuem], …, [paramn_value1, paramn_value2, …, paramn_valuem]] If a ndarray array is provided, with the same assumptions, it should be a 2-D array with shape (n, m) with the parameters values.

  • forward (bool or None, optional) – If True, search only in the forward continuation. If False, search only in the backward continuation. If None, search in both backward and forward direction. Default to None.

  • tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the values comparison if the selection rule based on parameters values is activated (by setting the arguments parameters and values). 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 of the number of parameters, each value in the array corresponding to a parameter. Default to 0.01.

Returns:

A list of the sought solutions

Return type:

list(AUTOSolution objects)

get_solution_by_index(idx)[source]

Get the AUTO solution object corresponding to a index.

Parameters:

idx (int) – The index of the point. Positive integers identify points on the forward continuation, while negative integers identify points on the backward continuation. The zero index identifies the initial point of the continuations.

Returns:

The sought AUTO solution object.

Return type:

AUTOSolution object

get_solution_by_label(label)[source]

Get the AUTO solution object corresponding to a given label.

Parameters:

label (str) – A string with the label of the point for which to return the solution object. Backward continuation label must be preceded by a ‘-’ character. E.g. ‘HB2’ identifies the second Hopf bifurcation point of the forward continuation, while ‘-UZ3’ identifies the third user-defined label of the backward branch.

Returns:

The sought AUTO solution object.

Return type:

AUTOSolution object

property isfixedpoint

True if the current object is a fixed point continuation.

Type:

bool

property isperiodicorbit

True if the current object is a periodic orbit continuation.

Type:

bool

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

Load the AUTO files and the branch parameters and metadata for both the forward and backward continuations if they exist.

Parameters:

  • filename (str) – The filename used to store the continuation(s) parameters and metadata to disk in Pickle format.

  • load_initial_data (bool) – Try or not to load the initial data field. Default to True.

abstract make_backward_continuation(initial_data, auto_suffix='', **continuation_kwargs)[source]

Make the backward continuation. To be implemented in subclasses.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.

abstract make_continuation(initial_data, auto_suffix='', only_forward=False, **continuation_kwargs)[source]

Make both forward and backward (if possible) continuation. To be implemented in subclasses.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • 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) – Keyword arguments to be passed to the AUTO continuation. See below for further details.

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. We provide below the most important ones:

Parameters:

  • DS (float, optional) – AUTO uses pseudo-arclength continuation for following solution families. The pseudo-arclength stepsize is the distance between the current solution and the next solution on a family. By default, this distance includes all state variables (or state functions) and all free parameters. The constant DS defines the pseudo-arclength stepsize to be used for the first attempted step along any family. DS may be chosen positive or negative; changing its sign reverses the direction of computation. The relation DSMIN ≤ | DS | ≤ DSMAX must be satisfied. The precise choice of DS is problem-dependent.

  • DSMIN (float, optional) – This is minimum allowable absolute value of the pseudo-arclength stepsize. DSMIN must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMIN is highly problem-dependent.

  • DSMAX (float, optional) – The maximum allowable absolute value of the pseudo-arclength stepsize. DSMAX must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMAX is highly problem-dependent.

  • NMX (int, optional) – The maximum number of steps to be taken along the branch.

  • IBR (int, optional) – This constant specifies the initial branch number BR that is used. The default IBR=0 means that that this number is determined automatically.

  • ILP (int, optional) –

    • If ILP=0: No detection of folds. This is the recommended choice in the AUTO documentation.

    • If ILP=1: Detection of folds. To be used if subsequent fold continuation is intended.

  • SP (list(str), optional) –

    This constant controls the detection of bifurcations and adds stopping conditions. It is specified as a list of bifurcation type strings followed by an optional number. If this number is 0, then the detection of this bifurcation is turned off, and if it is missing then the detection is turned on. A number n greater than zero specifies that the continuation should stop as soon as the nth bifurcation of this type has been reached. Examples:

    • SP=[’LP0’] turn off detection of folds.

    • SP=[’LP’,’HB3’,’BP0’,’UZ3’] turn on the detection of folds and Hopf bifurcations, turn off detection of branch points and stop at the third Hopf bifurcation or third user defined point, whichever comes first.

  • ISP (int, optional) –

    This constant controls the detection of Hopf bifurcations, branch points, period-doubling bifurcations, and torus bifurcations:

    • If ISP=0 This setting disables the detection of Hopf bifurcations, branch points, period doubling bifurcations, and torus bifurcations and the computation of Floquet multipliers.

    • If ISP=1 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points, period-doubling bifurcations and torus bifurcations are not detected for periodic solutions and boundary value problems. However, Floquet multipliers are computed.

    • If ISP=2 This setting enables the detection of all special solutions. For periodic solutions and rotations, the choice ISP=2 should be used with care, due to potential inaccuracy in the computation of the linearized Poincar ́e map and possible rapid variation of the Floquet multipliers.

    • If ISP=3 Hopf bifurcations will not be detected. Branch points will be detected, and AUTO will monitor the Floquet multipliers. Period-doubling and torus bifurcations will go undetected. This option is useful for certain problems with non-generic Floquet behavior.

    • If ISP=4 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points are not detected for periodic solutions and boundary value problems. AUTO will monitor the Floquet multipliers, and period-doubling and torus bifurcations will be detected.

  • ISW (int, optional) –

    This constant controls branch switching at branch points for the case of differential equations. Note that branch switching is automatic for algebraic equations.

    • If ISW=1 This is the normal value of ISW.

    • If ISW=-1 If IRS is the label of a branch point or a period-doubling bifurcation then branch switching will be done. For period doubling bifurcations it is recommended that NTST be increased.

    • If ISW=2 If IRS is the label of a fold, a Hopf bifurcation point, a period-doubling, a torus bifurcation, or, in a non-generic (symmetric) system, a branch point then a locus of such points will be computed.

    • If ISW=3 If IRS is the label of a branch point in a generic (non-symmetric) system then a locus of such points will be computed. Two additional free parameters must be specified for such continuations

  • MXBF (int, optional) – This constant, which is effective for algebraic problems only, sets the maximum number of bifurcations to be treated. Additional branch points will be noted, but the corresponding bifurcating families will not be computed.

  • dat (str, optional) – This constant, where dat=’filename’, sets the name of a user-supplied ASCII data file filename.dat, from which the continuation is to be restarted. The first column in the data file denotes the time, which does not need to be rescaled to the interval [0, 1], and further columns the coordinates of the solution. The parameter IRS must be set to 0.

  • PAR (dict, optional) – Determines the parameter values for the continuation to start from. Should be entred as {‘parameter_name’: parameter_value}.

  • IRS (int or str, optional) – This constant sets the label of the solution where the computation is to be restarted. Setting IRS=0 is typically used in the first run of a new problem. To restart the computation at the n-th label, use IRS=n. To restart the computation at a specific label (for example HB12), use IRS=HB12.

  • TY (str, optional) – This constant modifies the type from the restart solution. This is sometimes useful in conservative or extended systems, declaring a regular point to be a Hopf bifurcation point TY=’HB’ or a branch point TY=’BP’.

  • IPS (int, optional) –

    This constant defines the problem type:

    • If IPS=0 algebraic bifurcation problem.

    • If IPS=1 stationary solutions of ODEs with detection of Hopf bifurcations.

    • If IPS=-1 fixed points of the discrete dynamical systems.

    • If IPS=-2 time integration using implicit Euler.

    • If IPS=2 computation of periodic solutions.

    • If IPS=4 boundary value problems.

    • If IPS=5 algebraic optimization problems.

    • If IPS=7 boundary value problem with computation of Floquet multipliers.

    • If IPS=9 option is used in connection with the HomCont algorithms

    • If IPS=11 spatially uniform solutions of a system of parabolic PDEs, with detection of traveling wave bifurcations.

    • If IPS=12 continuation of traveling wave solutions to a system of parabolic PDEs.

    • If IPS=14 time evolution for a system of parabolic PDEs subject to periodic boundary conditions.

abstract make_forward_continuation(initial_data, auto_suffix='', **continuation_kwargs)[source]

Make the forward continuation. To be implemented in subclasses.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.

property number_of_points

The number of points of both forward and backward continuations.

Type:

dict(int)

property number_of_solutions

The number of solutions of both forward and backward continuations.

Type:

dict(int)

property path_name

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

Type:

str

property phase_space_variables

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

Type:

list(str)

plot_branch_parts(variables=(0, 1), ax=None, figsize=(10, 8), markersize=12.0, plot_kwargs=None, marker_kwargs=None, excluded_labels=('UZ', 'EP'), plot_sol_points=False, variables_name=None, cmap=None)[source]

Plots a bifurcation diagram along specified variables, with branches and the bifurcation points of fixed points and periodic orbits.

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 available_variables(). 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).

  • markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12.

  • plot_kwargs (dict or None, optional) – Optional key word arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.

  • marker_kwargs (dict or None, optional) – Optional key word 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.

  • cmap (cmap or None, optional) – The cmap used for plotting. If None, defaults to ‘Reds’

Returns:

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

Return type:

matplotlib.axes.Axes

plot_branch_parts_3D(variables=(0, 1, 2), ax=None, figsize=(10, 8), markersize=12.0, plot_kwargs=None, marker_kwargs=None, excluded_labels=('UZ', 'EP'), variables_name=None)[source]

Plots a bifurcation diagram on a 3-dimensional figure along specified variables, with branches and the bifurcation points of fixed points and periodic orbits.

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 available_variables(). The first value in the tuple determines the variable plot on the x-axis, the second the y-axis, and the third 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).

  • markersize (float, optional) – Size of the text plotted to display the bifurcation points. Defaults to 12..

  • plot_kwargs (dict or None, optional) – Optional key word arguments to pass to the plotting function, controlling the curves of the bifurcation diagram.

  • marker_kwargs (dict or None, optional) – Optional key word 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’].

  • 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 bifurcation diagram using a 3-dimensional projection.

Return type:

matplotlib.axes.Axes

plot_solutions(variables=(0, 1), ax=None, figsize=(10, 8), markersize=None, marker=None, linestyle=None, linewidth=None, color_solutions=False, plot_kwargs=None, labels=None, indices=None, parameter=None, value=None, variables_name=None, tol=0.01)[source]

Plots the stored solutions for a given set of variables. Fixed points are plotted using points, periodic orbits are plot as curves. Solutions that are plotted are filtered here using the selection rule-based get_filtered_solutions_list() method. However, if no selection rule is provided, it will plot all the solutions.

Parameters:

  • variables (tuple(int or str, int or str), optional) – The index or label of the variables shown in the 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).

  • ax (matplotlib.axes.Axes or None, optional) – A Matplotlib axis object to plot on. If None, a new figure is created.

  • figsize (figsize: tuple(float, float), optional) – Dimension of the figure that is returned, only used if ax is not passed. Defaults to (10, 8).

  • 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.

  • labels (str or list(str), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Label or list of labels to look for the solutions. Labels are two-characters string specifying the solutions types to return. For example ‘BP’ will return all the branching points of the continuations. This activates the selection rule based on labels and disable the others.

  • indices (int or list(int), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Index or list of indices to look for the solutions. AUTO index of solutions can be inquired for example by calling the print_summary() method of a given Continuation object. This activates the selection rule based on indices and disable the others.

  • parameter (str or list(str) or ndarray(str), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Parameter or list of parameters for which to match the solutions values to one provided by the values argument.

  • value (float or list(float) or ndarray(float), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). List of parameters values to match to the solutions parameters values. Can be a float if only one parameter is specified in the parameters arguments, otherwise a list or a ndarray array must be provided. If a list is provided, assuming n parameters were specified in the parameters argument and`m` values are sought, it should be a nested list with the following structure: [[param1_value1, param1_value2, …, param1_valuem], …, [paramn_value1, paramn_value2, …, paramn_valuem]] If a ndarray array is provided, with the same assumptions, it should be a 2-D array with shape (n, m) with the parameters values.

  • 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.

  • tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the values comparison if the selection rule based on parameters values is activated (by setting the arguments parameters and values). 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 of the number of parameters, each value in the array corresponding to a parameter. Default to 0.01.

Returns:

A Matplotlib axis object with the plot of the solutions.

Return type:

matplotlib.axes.Axes

plot_solutions_3D(variables=(0, 1, 2), ax=None, figsize=(10, 8), markersize=None, marker=None, linestyle=None, linewidth=None, plot_kwargs=None, labels=None, indices=None, parameter=None, value=None, variables_name=None, tol=0.01)[source]

Plots the stored solutions for a given set of variables, using a 3-dimensional projection. Fixed points are plotted using points, periodic orbits are plotted as curves. Solutions that are plotted are filtered here using the selection rule-based get_filtered_solutions_list() method. However, if no selection rule is provided, it will plot all the solutions.

Parameters:

  • variables (tuple(int or str, int or str, int or str), optional) – The index or label of the variables shown in the 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 three variables: (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).

  • 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.

  • labels (str or list(str), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Label or list of labels to look for the solutions. Labels are two-characters string specifying the solutions types to return. For example ‘BP’ will return all the branching points of the continuations. This activates the selection rule based on labels and disable the others.

  • indices (int or list(int), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Index or list of indices to look for the solutions. AUTO index of solutions can be inquired for example by calling the print_summary() method of a given Continuation object. This activates the selection rule based on indices and disable the others.

  • parameter (str or list(str) or ndarray(str), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). Parameter or list of parameters for which to match the solutions values to one provided by the values argument.

  • value (float or list(float) or ndarray(float), optional) – Selection rule to select particular solutions (see get_filtered_solutions_list() for more details.). List of parameters values to match to the solutions parameters values. Can be a float if only one parameter is specified in the parameters arguments, otherwise a list or a ndarray array must be provided. If a list is provided, assuming n parameters were specified in the parameters argument and`m` values are sought, it should be a nested list with the following structure: [[param1_value1, param1_value2, …, param1_valuem], …, [paramn_value1, paramn_value2, …, paramn_valuem]] If a ndarray array is provided, with the same assumptions, it should be a 2-D array with shape (n, m) with the parameters values.

  • 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.

  • tol (float or list(float) or ndarray(float), optional) – The numerical tolerance of the values comparison if the selection rule based on parameters values is activated (by setting the arguments parameters and values). 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 of the number of parameters, each value in the array corresponding to a parameter. Default to 0.01.

Returns:

A Matplotlib axis object with the 3-dimensional plot of the solutions.

Return type:

matplotlib.axes.Axes

point_diagnostic(idx)[source]

Method which returns the diagnostic of a given point of the continuation.

Parameters:

idx (str or int) –

AUTO index of the point to give the diagnostic from. If idx:

  • is a string, it should identify the label of the point, assuming the point has a label. Backward continuation label must be preceded by a ‘-’ character. E.g. ‘HB2’ would identify the second Hopf bifurcation point of the forward continuation, while ‘-UZ3’ identifies the third user-defined label of the backward branch.

  • is an integer, it corresponds to the index of the point. Positive integers identify points on the forward continuation, while negative integers identify points on the backward continuation. The zero index identifies the initial point of the continuations.

Returns:

The point diagnostic.

Return type:

str

print_diagnostics()[source]

Method which prints the full AUTO diagnostics of the continuations.

print_point_diagnostic(idx)[source]

Method which prints the diagnostic of a given point of the continuation.

Parameters:

idx (str or int) –

AUTO index of the point to give the diagnostic from. If idx:

  • is a string, it should identify the label of the point, assuming the point has a label. Backward continuation label must be preceded by a ‘-’ character. E.g. ‘HB2’ would identify the second Hopf bifurcation point of the forward continuation, while ‘-UZ3’ identifies the third user-defined label of the backward branch.

  • is an integer, it corresponds to the index of the point. Positive integers identify points on the forward continuation, while negative integers identify points on the backward continuation. The zero index identifies the initial point of the continuations.

print_summary()[source]

Print the summary of the AUTO continuations.

same_solutions_as(other, parameters, solutions_types=('HB', 'BP', 'UZ', 'PD', 'TR', 'LP'), tol=0.02)[source]

Check if the continuation has exactly the same solutions as another Continuation object.

Parameters:

  • other (Continuation) – The other continuation object to check the solutions.

  • parameters (list(str)) – The parameters to be considered to check if the solutions are the same.

  • solutions_types (list(str)) – The types of solution to consider in the search. Default to [‘HB’, ‘BP’, ‘UZ’, ‘PD’, ‘TR’, ‘LP’].

  • tol (float or list(float) or ndarray(float)) – The numerical tolerance of the comparison to determine if two solutions are equal. 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 of the number of parameters, each value in the array corresponding to a parameter.

Returns:

True if both continuations have exactly the same solutions according to the specified parameters.

Return type:

bool

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

Save the AUTO files and the branch parameters and metadata for both the forward and backward continuations if they exist. The branch parameters and metadata will be stored in a Pickle file.

Parameters:

  • filename (str or None, optional) – The filename used to store the continuation(s) parameters and metadata to disk in Pickle format. If None, will assign a default generic name. Default is None.

  • auto_filename_suffix (str or None, optional) – Suffix to use for the AUTO files used to save the continuation data from disk. If None, will assign a default generic suffix. Default is None.

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.

solutions_in(other, parameters, solutions_types=('HB', 'BP', 'UZ', 'PD', 'TR', 'LP'), tol=0.02, return_parameters=False, return_solutions=False)[source]

Check if the continuation solutions are all included in the solutions of another Continuation object.

Parameters:

  • other (Continuation) – The other continuation object to check the solutions.

  • parameters (list(str)) – The parameters to be considered to check if the solutions are the same.

  • solutions_types (list(str)) – The types of solution to consider in the search. Default to [‘HB’, ‘BP’, ‘UZ’, ‘PD’, ‘TR’, ‘LP’].

  • tol (float or list(float) or ndarray(float)) – The numerical tolerance of the comparison to determine if two solutions are equal. 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 of the number of parameters, each value in the array corresponding to a parameter.

  • return_parameters (bool) – If True, return an additional array with the values of the parameters for which the solutions of the two Continuation objects match.

Returns:

  • solutions_in (bool) – True if the solutions are all contained in the other continuations according to the specified parameters.

  • solutions_in_values (~numpy.ndarray:) – If return_parameters is True, an array with the values of the parameters for which the solutions of the two continuations match.

property solutions_index

The indices of all the solutions of both forward and backward continuations.

Type:

dict(list(int))

property solutions_label

The labels of all the solutions of both forward and backward continuations.

Type:

dict(list(str))

property solutions_list_by_direction

The full list of solutions sorted by direction for both forward and backward continuations.

Type:

dict(list(AUTOSolution object))

property solutions_list_by_direction_and_by_label

The full list of solutions sorted by labels and direction for both forward and backward continuations.

Type:

dict(list(AUTOSolution object))

solutions_parameters(parameters, solutions_types=('HB', 'BP', 'UZ', 'PD', 'TR', 'LP'), forward=None)[source]

Return the parameters value for the solutions of the backward and forward continuations.

Parameters:

  • parameters (list(str)) – The parameters to be returned.

  • solutions_types (list(str), optional) – The types of solution to consider in the search. Default to [‘HB’, ‘BP’, ‘UZ’, ‘PD’, ‘TR’, ‘LP’].

  • forward (bool or None, optional) – If True, search only in the forward continuation. If False, search only in the backward continuation. If None, search in both backward and forward direction. Default to None.

Returns:

The values of the requested solutions parameters.

Return type:

ndarray

solutions_part_of(other, parameters, solutions_types=('HB', 'BP', 'UZ', 'PD', 'TR', 'LP'), tol=0.02, return_parameters=False, return_solutions=False, forward=None)[source]

Check if a subset of the continuation solutions are included in the solutions of another Continuation object.

Parameters:

  • other (Continuation) – The other continuation object to check the solutions.

  • parameters (list(str)) – The parameters to be considered to check if the solutions are the same.

  • solutions_types (list(str)) – The types of solution to consider in the search. Default to [‘HB’, ‘BP’, ‘UZ’, ‘PD’, ‘TR’, ‘LP’].

  • tol (float or list(float) or ndarray(float)) – The numerical tolerance of the comparison to determine if two solutions are equal. 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 of the number of parameters, each value in the array corresponding to a parameter.

  • return_parameters (bool) – If True, return an additional array with the values of the parameters for which the solutions of the two Continuation objects match.

  • return_solutions (bool) – If True, return an additional list with the AUTO solution objects.

  • forward (bool or None, optional) – If True, search only in the forward continuation. If False, search only in the backward continuation. If None, search in both backward and forward direction. Default to None.

Returns:

  • part_of (bool) – True if the solutions are all contained in the other continuations according to the specified parameters.

  • solutions_part_of_values (~numpy.ndarray:) – If return_parameters is True, an array with the values of the parameters for which the solutions of the two continuations match.

  • solutions_part_of (list(AUTOSolution object):) – If return_solutions is True, a list of the solutions which are included in the other Continuation object.

property stability

The stability information of both forward and backward continuations.

Type:

dict(list)

summary()[source]

Return the summary of the AUTO continuations.

Returns:

The summary.

Return type:

str

temporary_chdir(new_dir)[source]

As AUTOCommands does not provide functionality to pass a filepath, this is a workaround to switch paths for loading/saving.

Fixed points continuation module

Fixed Point Continuation class definition

This module implements the fixed point continuation methods.

class auto2.continuations.fixed_points.FixedPointContinuation(model_name, config_object, path_name=None)[source]

Bases: Continuation

Class for the fixed point continuations in auto-AUTO.

Parameters:

  • model_name (str) – The name of the model to load. Used to load the .f90 file of the provided model.

  • config_object (ConfigParser) – A loaded ConfigParser object.

  • path_name (str, optional) – The directory path where files are read/saved. If None, defaults to current working directory.

model_name

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

Type:

str

config_object

A loaded ConfigParser object.

Type:

ConfigParser

continuation

Dictionary holding the forward and backward continuation data.

Type:

dict

branch_number

The AUTO branch number attributed to the continuation(s). This number can be set manually by passing the IBR AUTO parameter when starting the continuations (see the make_continuation() documentation for more details).

Type:

int

initial_data

The initial data used to start the continuation(s).

Type:

ndarray or AUTO solution object

auto_filename_suffix

Suffix for the AUTO files used to save the continuation(s) data and parameters on disk.

Type:

str

property isfixedpoint

Whether this class stores fixed point continuations.

Type:

bool

property isperiodicorbit

Whether this class stores periodic orbit continuations.

Type:

bool

make_backward_continuation(initial_data, auto_suffix='', **continuation_kwargs)[source]

Make the backward continuation of fixed points.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.

make_continuation(initial_data, auto_suffix='', only_forward=False, **continuation_kwargs)[source]

Make both forward and backward (if possible) continuation of fixed points.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • 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) – Keyword arguments to be passed to the AUTO continuation. See below for further details.

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. We provide below the most important ones:

Parameters:

  • DS (float, optional) – AUTO uses pseudo-arclength continuation for following solution families. The pseudo-arclength stepsize is the distance between the current solution and the next solution on a family. By default, this distance includes all state variables (or state functions) and all free parameters. The constant DS defines the pseudo-arclength stepsize to be used for the first attempted step along any family. DS may be chosen positive or negative; changing its sign reverses the direction of computation. The relation DSMIN ≤ | DS | ≤ DSMAX must be satisfied. The precise choice of DS is problem-dependent.

  • DSMIN (float, optional) – This is minimum allowable absolute value of the pseudo-arclength stepsize. DSMIN must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMIN is highly problem-dependent.

  • DSMAX (float, optional) – The maximum allowable absolute value of the pseudo-arclength stepsize. DSMAX must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMAX is highly problem-dependent.

  • NMX (int, optional) – The maximum number of steps to be taken along the branch.

  • IBR (int, optional) – This constant specifies the initial branch number BR that is used. The default IBR=0 means that that this number is determined automatically.

  • ILP (int, optional) –

    • If ILP=0: No detection of folds. This is the recommended choice in the AUTO documentation.

    • If ILP=1: Detection of folds. To be used if subsequent fold continuation is intended.

  • SP (list(str), optional) –

    This constant controls the detection of bifurcations and adds stopping conditions. It is specified as a list of bifurcation type strings followed by an optional number. If this number is 0, then the detection of this bifurcation is turned off, and if it is missing then the detection is turned on. A number n greater than zero specifies that the continuation should stop as soon as the nth bifurcation of this type has been reached. Examples:

    • SP=[’LP0’] turn off detection of folds.

    • SP=[’LP’,’HB3’,’BP0’,’UZ3’] turn on the detection of folds and Hopf bifurcations, turn off detection of branch points and stop at the third Hopf bifurcation or third user defined point, whichever comes first.

  • ISP (int, optional) –

    This constant controls the detection of Hopf bifurcations, branch points, period-doubling bifurcations, and torus bifurcations:

    • If ISP=0 This setting disables the detection of Hopf bifurcations, branch points, period doubling bifurcations, and torus bifurcations and the computation of Floquet multipliers.

    • If ISP=1 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points, period-doubling bifurcations and torus bifurcations are not detected for periodic solutions and boundary value problems. However, Floquet multipliers are computed.

    • If ISP=2 This setting enables the detection of all special solutions. For periodic solutions and rotations, the choice ISP=2 should be used with care, due to potential inaccuracy in the computation of the linearized Poincar ́e map and possible rapid variation of the Floquet multipliers.

    • If ISP=3 Hopf bifurcations will not be detected. Branch points will be detected, and AUTO will monitor the Floquet multipliers. Period-doubling and torus bifurcations will go undetected. This option is useful for certain problems with non-generic Floquet behavior.

    • If ISP=4 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points are not detected for periodic solutions and boundary value problems. AUTO will monitor the Floquet multipliers, and period-doubling and torus bifurcations will be detected.

  • ISW (int, optional) –

    This constant controls branch switching at branch points for the case of differential equations. Note that branch switching is automatic for algebraic equations.

    • If ISW=1 This is the normal value of ISW.

    • If ISW=-1 If IRS is the label of a branch point or a period-doubling bifurcation then branch switching will be done. For period doubling bifurcations it is recommended that NTST be increased.

    • If ISW=2 If IRS is the label of a fold, a Hopf bifurcation point, a period-doubling, a torus bifurcation, or, in a non-generic (symmetric) system, a branch point then a locus of such points will be computed.

    • If ISW=3 If IRS is the label of a branch point in a generic (non-symmetric) system then a locus of such points will be computed. Two additional free parameters must be specified for such continuations

  • MXBF (int, optional) – This constant, which is effective for algebraic problems only, sets the maximum number of bifurcations to be treated. Additional branch points will be noted, but the corresponding bifurcating families will not be computed.

  • dat (str, optional) – This constant, where dat=’filename’, sets the name of a user-supplied ASCII data file filename.dat, from which the continuation is to be restarted. The first column in the data file denotes the time, which does not need to be rescaled to the interval [0, 1], and further columns the coordinates of the solution. The parameter IRS must be set to 0.

  • PAR (dict, optional) – Determines the parameter values for the continuation to start from. Should be entred as {‘parameter_name’: parameter_value}.

  • IRS (int or str, optional) – This constant sets the label of the solution where the computation is to be restarted. Setting IRS=0 is typically used in the first run of a new problem. To restart the computation at the n-th label, use IRS=n. To restart the computation at a specific label (for example HB12), use IRS=HB12.

  • TY (str, optional) – This constant modifies the type from the restart solution. This is sometimes useful in conservative or extended systems, declaring a regular point to be a Hopf bifurcation point TY=’HB’ or a branch point TY=’BP’.

  • IPS (int, optional) –

    This constant defines the problem type:

    • If IPS=0 algebraic bifurcation problem.

    • If IPS=1 stationary solutions of ODEs with detection of Hopf bifurcations.

    • If IPS=-1 fixed points of the discrete dynamical systems.

    • If IPS=-2 time integration using implicit Euler.

    • If IPS=2 computation of periodic solutions.

    • If IPS=4 boundary value problems.

    • If IPS=5 algebraic optimization problems.

    • If IPS=7 boundary value problem with computation of Floquet multipliers.

    • If IPS=9 option is used in connection with the HomCont algorithms

    • If IPS=11 spatially uniform solutions of a system of parabolic PDEs, with detection of traveling wave bifurcations.

    • If IPS=12 continuation of traveling wave solutions to a system of parabolic PDEs.

    • If IPS=14 time evolution for a system of parabolic PDEs subject to periodic boundary conditions.

make_forward_continuation(initial_data, auto_suffix='', **continuation_kwargs)[source]

Make the forward continuation of fixed points.

Parameters:

  • initial_data (ndarray or AUTOSolution) – Initial data used to start the continuation(s).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.

point_stability(idx)[source]

Given a point index, returns the stability values (Eiganvalues) for this point.

Parameters:

idx (str, int) –

AUTO index of the point to give the diagnostic from. If idx:

  • is a string, it should identify the label of the point, assuming the point has a label. Backward continuation label must be preceded by a ‘-’ character. E.g. ‘HB2’ would identify the second Hopf bifurcation point of the forward continuation, while ‘-UZ3’ identifies the third user-defined label of the backward branch.

  • is an integer, it corresponds to the index of the point. Positive integers identify points on the forward continuation, while negative integers identify points on the backward continuation. The zero index identifies the initial point of the continuations.

Return type:

ndarray

Periodic orbits continuation module

Periodic Orbit Continuation class definition

This module implements the periodic orbit continuation methods.

class auto2.continuations.periodic_orbits.PeriodicOrbitContinuation(model_name, config_object, path_name=None)[source]

Bases: Continuation

Class for the periodic orbit continuations in auto-AUTO.

Parameters:

  • model_name (str) – The name of the model to load. Used to load the .f90 file of the provided model.

  • config_object (ConfigParser) – A loaded ConfigParser object.

  • path_name (str, optional) – The directory path where files are read/saved. If None, defaults to current working directory.

model_name

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

Type:

str

config_object

A loaded ConfigParser object.

Type:

ConfigParser

continuation

Dictionary holding the forward and backward continuation data.

Type:

dict

branch_number

The AUTO branch number attributed to the continuation(s). This number can be set manually by passing the IBR AUTO parameter when starting the continuations (see the make_continuation() documentation for more details).

Type:

int

initial_data

The initial data used to start the continuation(s).

Type:

ndarray or AUTO solution object

auto_filename_suffix

Suffix for the AUTO files used to save the continuation(s) data and parameters on disk.

Type:

str

property isfixedpoint

True if the current object is a fixed point continuation.

Type:

bool

property isperiodicorbit

True if the current object is a periodic orbit continuation.

Type:

bool

make_backward_continuation(initial_data, auto_suffix='', max_bp=None, **continuation_kwargs)[source]

Make the backward continuation of periodic orbits.

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 in the make_continuation() documentation).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • max_bp (int or None) – Number of branching points detected before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.

make_continuation(initial_data, auto_suffix='', only_forward=True, max_bp=None, **continuation_kwargs)[source]

Make both forward and backward (if possible) continuation for fixed points.

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).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • 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.

  • max_bp (int or None) – Number of branching points detected before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details.

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. We provide below the most important ones:

Parameters:

  • DS (float, optional) – AUTO uses pseudo-arclength continuation for following solution families. The pseudo-arclength stepsize is the distance between the current solution and the next solution on a family. By default, this distance includes all state variables (or state functions) and all free parameters. The constant DS defines the pseudo-arclength stepsize to be used for the first attempted step along any family. DS may be chosen positive or negative; changing its sign reverses the direction of computation. The relation DSMIN ≤ | DS | ≤ DSMAX must be satisfied. The precise choice of DS is problem-dependent.

  • DSMIN (float, optional) – This is minimum allowable absolute value of the pseudo-arclength stepsize. DSMIN must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMIN is highly problem-dependent.

  • DSMAX (float, optional) – The maximum allowable absolute value of the pseudo-arclength stepsize. DSMAX must be positive. It is only effective if the pseudo-arclength step is adaptive, i.e., if IADS>0. The choice of DSMAX is highly problem-dependent.

  • NMX (int, optional) – The maximum number of steps to be taken along the branch.

  • IBR (int, optional) – This constant specifies the initial branch number BR that is used. The default IBR=0 means that that this number is determined automatically.

  • ILP (int, optional) –

    • If ILP=0: No detection of folds. This is the recommended choice in the AUTO documentation.

    • If ILP=1: Detection of folds. To be used if subsequent fold continuation is intended.

  • SP (list(str), optional) –

    This constant controls the detection of bifurcations and adds stopping conditions. It is specified as a list of bifurcation type strings followed by an optional number. If this number is 0, then the detection of this bifurcation is turned off, and if it is missing then the detection is turned on. A number n greater than zero specifies that the continuation should stop as soon as the nth bifurcation of this type has been reached. Examples:

    • SP=[’LP0’] turn off detection of folds.

    • SP=[’LP’,’HB3’,’BP0’,’UZ3’] turn on the detection of folds and Hopf bifurcations, turn off detection of branch points and stop at the third Hopf bifurcation or third user defined point, whichever comes first.

  • ISP (int, optional) –

    This constant controls the detection of Hopf bifurcations, branch points, period-doubling bifurcations, and torus bifurcations:

    • If ISP=0 This setting disables the detection of Hopf bifurcations, branch points, period doubling bifurcations, and torus bifurcations and the computation of Floquet multipliers.

    • If ISP=1 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points, period-doubling bifurcations and torus bifurcations are not detected for periodic solutions and boundary value problems. However, Floquet multipliers are computed.

    • If ISP=2 This setting enables the detection of all special solutions. For periodic solutions and rotations, the choice ISP=2 should be used with care, due to potential inaccuracy in the computation of the linearized Poincar ́e map and possible rapid variation of the Floquet multipliers.

    • If ISP=3 Hopf bifurcations will not be detected. Branch points will be detected, and AUTO will monitor the Floquet multipliers. Period-doubling and torus bifurcations will go undetected. This option is useful for certain problems with non-generic Floquet behavior.

    • If ISP=4 Branch points and Hopf bifurcations are detected for algebraic equations. Branch points are not detected for periodic solutions and boundary value problems. AUTO will monitor the Floquet multipliers, and period-doubling and torus bifurcations will be detected.

  • ISW (int, optional) –

    This constant controls branch switching at branch points for the case of differential equations. Note that branch switching is automatic for algebraic equations.

    • If ISW=1 This is the normal value of ISW.

    • If ISW=-1 If IRS is the label of a branch point or a period-doubling bifurcation then branch switching will be done. For period doubling bifurcations it is recommended that NTST be increased.

    • If ISW=2 If IRS is the label of a fold, a Hopf bifurcation point, a period-doubling, a torus bifurcation, or, in a non-generic (symmetric) system, a branch point then a locus of such points will be computed.

    • If ISW=3 If IRS is the label of a branch point in a generic (non-symmetric) system then a locus of such points will be computed. Two additional free parameters must be specified for such continuations

  • MXBF (int, optional) – This constant, which is effective for algebraic problems only, sets the maximum number of bifurcations to be treated. Additional branch points will be noted, but the corresponding bifurcating families will not be computed.

  • dat (str, optional) – This constant, where dat=’filename’, sets the name of a user-supplied ASCII data file filename.dat, from which the continuation is to be restarted. The first column in the data file denotes the time, which does not need to be rescaled to the interval [0, 1], and further columns the coordinates of the solution. The parameter IRS must be set to 0.

  • PAR (dict, optional) – Determines the parameter values for the continuation to start from. Should be entred as {‘parameter_name’: parameter_value}.

  • IRS (int or str, optional) – This constant sets the label of the solution where the computation is to be restarted. Setting IRS=0 is typically used in the first run of a new problem. To restart the computation at the n-th label, use IRS=n. To restart the computation at a specific label (for example HB12), use IRS=HB12.

  • TY (str, optional) – This constant modifies the type from the restart solution. This is sometimes useful in conservative or extended systems, declaring a regular point to be a Hopf bifurcation point TY=’HB’ or a branch point TY=’BP’.

  • IPS (int, optional) –

    This constant defines the problem type:

    • If IPS=0 algebraic bifurcation problem.

    • If IPS=1 stationary solutions of ODEs with detection of Hopf bifurcations.

    • If IPS=-1 fixed points of the discrete dynamical systems.

    • If IPS=-2 time integration using implicit Euler.

    • If IPS=2 computation of periodic solutions.

    • If IPS=4 boundary value problems.

    • If IPS=5 algebraic optimization problems.

    • If IPS=7 boundary value problem with computation of Floquet multipliers.

    • If IPS=9 option is used in connection with the HomCont algorithms

    • If IPS=11 spatially uniform solutions of a system of parabolic PDEs, with detection of traveling wave bifurcations.

    • If IPS=12 continuation of traveling wave solutions to a system of parabolic PDEs.

    • If IPS=14 time evolution for a system of parabolic PDEs subject to periodic boundary conditions.

make_forward_continuation(initial_data, auto_suffix='', max_bp=None, **continuation_kwargs)[source]

Make the forward continuation of periodic orbits.

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 in the make_continuation() documentation).

  • auto_suffix (str, optional) – Suffix to use for the AUTO and Pickle files used to save the continuation(s) data, parameters and metadata on disk. If not provided, does not save the data on disk.

  • max_bp (int or None) – Number of branching points detected before turning off the detection of branching points. If None, the detection of branching points is never turned off. Defaults to None.

  • continuation_kwargs (dict) – Keyword arguments to be passed to the AUTO continuation. See below for further details

Notes

AUTO Continuation Keyword Arguments: See Section 10.8 in the AUTO documentation for further details. The most important ones are provided in the documentation of the make_continuation() method.