"""
The bifurcation diagram
==========================
"""

#############################
# Link to the logistic map
# -----------------------------
# It is very easy to build the bifurcation diagram of the logistic map. You only need to iterate the logistic map long enough for a value
# of :math:`r` (:math:`x_{n+1}=rx_n(1-x_n)`) and then plot the last point. The goal of iterating the sequence a lot of time is to see where
# the points are attracted to.
# If you only need the values of the bifurcation diagram and not the actual plot, only pass the value False to the argument show.


import chaoseverywhere as chaos

chaos.bifurcation(show=True)


#################################
# An animated visualization
# -------------------------------
# The bifurcation diagram surprinsgly has a link to the Mandelbrot set. Indeed, there is a bijection between the growth ratio :math:`r` and
# value (real part) of :math:`c` in the Mandelbrot formula.
#
#
# .. raw:: html
#
#     <iframe width="560" height="315" src="https://www.youtube.com/embed/xYQbqML1eE4" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
#
#
# Another way to look at it is: if we plot the real part of the iterates of the points
# in the Mandelbrot set, then we get the bifurcation diagram.