Package PyDSTool :: Package Toolbox :: Module phaseplane :: Class nullcline

Class nullcline

object --+
         |
        nullcline

Nullcline representation class in 2D (x, y) plane parameterizable
by x variable only. A third-order univariate spline will be fitted to
the given sample point array using all as knots, and available
through the 'spline' attribute.

Input:
  Names are given to the x and y axis directions at initialization.
  Third argument nullc_array is the N-by-2 array of N two-dimensional
    data points.
  The final, optional argument x_relative_scale indicates the relative
    scale of the x values' domain compared to the y values. This is
    used to provide a form of subjective normalization for measuring
    curvature and other geometric properties when plotting y vs. x
    on these scales. E.g if x values range over -50 to +50, while
    y ranges over 0 to 1, x_relative_scale=100 is appropriate.

N.B.: Only works for nullclines that can be written y = f(x). Future
versions may support non-functions.

Some functions acting on nullclines currently also rely on monotonicity
of y=f(x), and will verify this using the is_monotonic method before
proceeding. Future versions will attempt to resolve that.

Instance Methods

[hide private]

__init__(self, xname, yname, nullc_array, x_relative_scale=1)
x.__init__(...) initializes x; see x.__class__.__doc__ for signature

source code

__call__(self, x)
Returns a scalar if x is scalar.

source code

createParameterization(self)
Creates a (non-arc-length) parameterization

source code

toPointset(self, with_param=False)

source code

sample_x_domain(self, refine=0)

source code

tgt_vec(self, x, rescale=False)
Return tangent vector to an interior point of nullcline, normalized to length 1.

source code

curvature(self, xdata)
Signed curvature at x is returned (scalar) for scalar x, otherwise array of such scalars for 1D array x.

source code

grad_curvature(self, xdata)
Derivative of the curvature at x is returned (scalar) for scalar x, otherwise array of such scalars for 1D array x.

source code

curvature_at_sample_points(self, refine=0)
Returns array of signed scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

source code

grad_curvature_at_sample_points(self, refine=0)
Returns array of signed scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

source code

concavity(self, xdata)
Concavity scalar +/- 1 or 0 at x is returned for scalar x, otherwise array of such scalars for 1D array x.

source code

concavity_at_sample_points(self, refine=0)
Returns array of +/- 1 or 0 scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

source code

crop(self, xdom, ydom, resample_frac=0.1)
Returns a new Nullcline object cropped to the (x, y) domain given by pairs xdom and ydom.

source code

is_monotonic(self)

source code

__copy__(self)

source code

Inherited from object: __delattr__, __getattribute__, __hash__, __new__, __reduce__, __reduce_ex__, __repr__, __setattr__, __str__

Properties

[hide private]

Inherited from object: __class__

Method Details

[hide private]

init(self, xname, yname, nullc_array, x_relative_scale=1)
(Constructor)

source code

x.__init__(...) initializes x; see x.__class__.__doc__ for signature

Overrides: object.__init__: (inherited documentation)

tgt_vec(self, x, rescale=False)

source code

Return tangent vector to an interior point of nullcline, normalized to length 1. Set rescale option (default False) to return a vector rescaled according to relative x vs. y scales (thus it's no longer a tangent vector in that case).

curvature(self, xdata)

source code

Signed curvature at x is returned (scalar) for scalar x, otherwise array of such scalars for 1D array x.

Positive values mean concave "up" in the plane.

curvature_at_sample_points(self, refine=0)

source code

Returns array of signed scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

Positive values mean concave "up" in the plane. Assumes nullcline is monotonic.

grad_curvature_at_sample_points(self, refine=0)

source code

Returns array of signed scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

Assumes nullcline is monotonic.

concavity(self, xdata)

source code

Concavity scalar +/- 1 or 0 at x is returned for scalar x, otherwise array of such scalars for 1D array x.

Positive values mean concave "up" in the plane.

concavity_at_sample_points(self, refine=0)

source code

Returns array of +/- 1 or 0 scalars for each x data point (knot) of the nullcline spline (computed just inside endpoints by 1% of distance to next innermost sample point).

Positive values mean concave "up" in the plane. Assumes nullcline is monotonic.

crop(self, xdom, ydom, resample_frac=0.1)

source code

Returns a new Nullcline object cropped to the (x, y) domain given by pairs xdom and ydom.

If resample_frac > 0 (default 0.1), points at the domain ends and additional sample points will be included in the new nullcline object regardless of whether they were in the original sample point set for the cropped nullcline. New points will be selected at the rate indicated, measured as a fraction of the domain width. Points that are as close as 1/10th of this rate to existing sample points will be excluded, to avoid ill-conditioning the spline shape in the resampling.

Given the assumption that nullclines are montonic functions of x, this guarantees a unique, contiguous piece of the nullcline is returned.

Class nullcline

__init__(self, xname, yname, nullc_array, x_relative_scale=1) (Constructor)

tgt_vec(self, x, rescale=False)

curvature(self, xdata)

curvature_at_sample_points(self, refine=0)

grad_curvature_at_sample_points(self, refine=0)

concavity(self, xdata)

concavity_at_sample_points(self, refine=0)

crop(self, xdom, ydom, resample_frac=0.1)

init(self, xname, yname, nullc_array, x_relative_scale=1)
(Constructor)