julia_sets.py

"""Author Alexandre De Zotti

Draws Julia sets of quadratic polynomials and exponential maps.
 More specifically, this iterates the function a fixed number of times
 then plots whether the absolute value of the last iterate is greater than
 a fixed threshold (named "escape radius"). For the exponential map this is not
 really an escape radius but rather a convenient way to approximate the Julia
 set with bounded orbits.

The examples presented here are:
- The Cauliflower Julia set, see e.g.
https://en.wikipedia.org/wiki/File:Julia_z2%2B0,25.png
- Other examples from https://en.wikipedia.org/wiki/Julia_set
- An exponential map Julia set, ambiantly homeomorphic to the examples in
http://www.math.univ-toulouse.fr/~cheritat/GalII/galery.html
 and
https://ddd.uab.cat/pub/pubmat/02141493v43n1/02141493v43n1p27.pdf

Remark: Some overflow runtime warnings are suppressed. This is because of the
 way the iteration loop is implemented, using numpy's efficient computations.
 Overflows and infinites are replaced after each step by a large number.
"""

import warnings
from typing import Any, Callable

import numpy
from matplotlib import pyplot

c_cauliflower = 0.25 + 0.0j
c_polynomial_1 = -0.4 + 0.6j
c_polynomial_2 = -0.1 + 0.651j
c_exponential = -2.0
nb_iterations = 56
window_size = 2.0
nb_pixels = 666


def eval_exponential(c_parameter: complex, z_values: numpy.ndarray) -> numpy.ndarray:
    """
    Evaluate $e^z + c$.
    >>> eval_exponential(0, 0)
    1.0
    >>> abs(eval_exponential(1, numpy.pi*1.j)) < 1e-15
    True
    >>> abs(eval_exponential(1.j, 0)-1-1.j) < 1e-15
    True
    """
    return numpy.exp(z_values) + c_parameter


def eval_quadratic_polynomial(
    c_parameter: complex, z_values: numpy.ndarray
) -> numpy.ndarray:
    """
    >>> eval_quadratic_polynomial(0, 2)
    4
    >>> eval_quadratic_polynomial(-1, 1)
    0
    >>> round(eval_quadratic_polynomial(1.j, 0).imag)
    1
    >>> round(eval_quadratic_polynomial(1.j, 0).real)
    0
    """
    return z_values * z_values + c_parameter


def prepare_grid(window_size: float, nb_pixels: int) -> numpy.ndarray:
    """
    Create a grid of complex values of size nb_pixels*nb_pixels with real and
     imaginary parts ranging from -window_size to window_size (inclusive).
    Returns a numpy array.

    >>> prepare_grid(1,3)
    array([[-1.-1.j, -1.+0.j, -1.+1.j],
           [ 0.-1.j,  0.+0.j,  0.+1.j],
           [ 1.-1.j,  1.+0.j,  1.+1.j]])
    """
    x = numpy.linspace(-window_size, window_size, nb_pixels)
    x = x.reshape((nb_pixels, 1))
    y = numpy.linspace(-window_size, window_size, nb_pixels)
    y = y.reshape((1, nb_pixels))
    return x + 1.0j * y


def iterate_function(
    eval_function: Callable[[Any, numpy.ndarray], numpy.ndarray],
    function_params: Any,
    nb_iterations: int,
    z_0: numpy.ndarray,
    infinity: float = None,
) -> numpy.ndarray:
    """
    Iterate the function "eval_function" exactly nb_iterations times.
    The first argument of the function is a parameter which is contained in
    function_params. The variable z_0 is an array that contains the initial
    values to iterate from.
    This function returns the final iterates.

    >>> iterate_function(eval_quadratic_polynomial, 0, 3, numpy.array([0,1,2])).shape
    (3,)
    >>> numpy.round(iterate_function(eval_quadratic_polynomial,
    ... 0,
    ... 3,
    ... numpy.array([0,1,2]))[0])
    0j
    >>> numpy.round(iterate_function(eval_quadratic_polynomial,
    ... 0,
    ... 3,
    ... numpy.array([0,1,2]))[1])
    (1+0j)
    >>> numpy.round(iterate_function(eval_quadratic_polynomial,
    ... 0,
    ... 3,
    ... numpy.array([0,1,2]))[2])
    (256+0j)
    """

    z_n = z_0.astype("complex64")
    for i in range(nb_iterations):
        z_n = eval_function(function_params, z_n)
        if infinity is not None:
            numpy.nan_to_num(z_n, copy=False, nan=infinity)
            z_n[abs(z_n) == numpy.inf] = infinity
    return z_n


def show_results(
    function_label: str,
    function_params: Any,
    escape_radius: float,
    z_final: numpy.ndarray,
) -> None:
    """
    Plots of whether the absolute value of z_final is greater than
    the value of escape_radius. Adds the function_label and function_params to
    the title.

    >>> show_results('80', 0, 1, numpy.array([[0,1,.5],[.4,2,1.1],[.2,1,1.3]]))
    """

    abs_z_final = (abs(z_final)).transpose()
    abs_z_final[:, :] = abs_z_final[::-1, :]
    pyplot.matshow(abs_z_final < escape_radius)
    pyplot.title(f"Julia set of ${function_label}$, $c={function_params}$")
    pyplot.show()


def ignore_overflow_warnings() -> None:
    """
    Ignore some overflow and invalid value warnings.

    >>> ignore_overflow_warnings()
    """
    warnings.filterwarnings(
        "ignore", category=RuntimeWarning, message="overflow encountered in multiply"
    )
    warnings.filterwarnings(
        "ignore",
        category=RuntimeWarning,
        message="invalid value encountered in multiply",
    )
    warnings.filterwarnings(
        "ignore", category=RuntimeWarning, message="overflow encountered in absolute"
    )
    warnings.filterwarnings(
        "ignore", category=RuntimeWarning, message="overflow encountered in exp"
    )


if __name__ == "__main__":

    z_0 = prepare_grid(window_size, nb_pixels)

    ignore_overflow_warnings()  # See file header for explanations

    nb_iterations = 24
    escape_radius = 2 * abs(c_cauliflower) + 1
    z_final = iterate_function(
        eval_quadratic_polynomial,
        c_cauliflower,
        nb_iterations,
        z_0,
        infinity=1.1 * escape_radius,
    )
    show_results("z^2+c", c_cauliflower, escape_radius, z_final)

    nb_iterations = 64
    escape_radius = 2 * abs(c_polynomial_1) + 1
    z_final = iterate_function(
        eval_quadratic_polynomial,
        c_polynomial_1,
        nb_iterations,
        z_0,
        infinity=1.1 * escape_radius,
    )
    show_results("z^2+c", c_polynomial_1, escape_radius, z_final)

    nb_iterations = 161
    escape_radius = 2 * abs(c_polynomial_2) + 1
    z_final = iterate_function(
        eval_quadratic_polynomial,
        c_polynomial_2,
        nb_iterations,
        z_0,
        infinity=1.1 * escape_radius,
    )
    show_results("z^2+c", c_polynomial_2, escape_radius, z_final)

    nb_iterations = 12
    escape_radius = 10000.0
    z_final = iterate_function(
        eval_exponential,
        c_exponential,
        nb_iterations,
        z_0 + 2,
        infinity=1.0e10,
    )
    show_results("e^z+c", c_exponential, escape_radius, z_final)