r"""
Henon Map
=========

The Henon map is a simple 2D quadratic map,

.. math::

    x_{n+1} = 1 - a x_n^2 + y_n, \qquad y_{n+1} = b x_n

whose classic parameters (:math:`a=1.4`, :math:`b=0.3`) produce a strange
attractor with the map's signature fractal, self-similar structure. Unlike
the (area-preserving) standard and baker's maps, the Henon map is
dissipative -- its Jacobian determinant is the constant :math:`-b`, with
:math:`|b|<1` -- so nearby points contract in area on average even while
they stretch apart along the attractor's unstable direction. This example
iterates a single long trajectory and scatter-plots it.
"""

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.maps import HenonMap
from physicskit.chaos.visualizers import theme
from physicskit.chaos.visualizers.bifurcation import map_bifurcation_sampler, plot_bifurcation_diagram

system = HenonMap(a=1.4, b=0.3)

# %%
# The finished attractor
# --------------------------
# Discard a short initial transient so the plotted points already lie on the
# attractor, then plot the full, much longer orbit at once to reveal the
# map's signature fractal, self-similar structure -- especially visible when
# zooming into the folds.
traj = system.trajectory(np.array([0.0, 0.0]), n_iter=50000)
traj = traj[100:]

fig, ax = plt.subplots(figsize=(7, 5))
ax.scatter(traj[:, 0], traj[:, 1], s=0.3, color=theme.PRIMARY, alpha=0.6)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Henon map attractor")

plt.show()

# %%
# The route there: a bifurcation diagram over a
# ---------------------------------------------------
# The single attractor above is only the end state at one fixed ``a``
# (``b`` held at its classic 0.3). Sweeping ``a`` and, for each value,
# plotting the surviving long-term ``x`` values -- exactly the same
# :func:`~physicskit.chaos.visualizers.bifurcation.plot_bifurcation_diagram`
# machinery used for the :doc:`Logistic Map </api/gallery/chaos/maps/plot_logistic_map>`
# -- shows *how* that attractor is reached: a period-doubling cascade out of
# simple fixed points and cycles, opening into the fractal chaotic band at
# the classic ``a = 1.4`` used above (dashed line), interrupted by visible
# periodic windows.
a_values = np.linspace(0.8, 1.42, 1200)
sampler = map_bifurcation_sampler(lambda a: HenonMap(a=a, b=0.3), state0=np.array([0.0, 0.0]), n_transient=300, n_keep=150)

fig2, ax2 = plot_bifurcation_diagram(a_values, sampler)
ax2.axvline(1.4, color=theme.MUTED, ls="--", lw=1.0, label="a = 1.4 (shown above)")
ax2.set_xlabel("a")
ax2.set_ylabel("x (long-term values)")
ax2.set_title("Henon map bifurcation diagram (b = 0.3)")
ax2.legend(fontsize=8)

plt.show()