.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/maps/plot_logistic_map.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_chaos_maps_plot_logistic_map.py: Logistic Map: the Period-Doubling Route to Chaos ==================================================== The logistic map, .. math:: x_{n+1} = r x_n (1 - x_n), is the simplest possible model of the period-doubling cascade: as the single growth-rate parameter :math:`r` is increased over :math:`[0, 4]`, a stable fixed point splits into a stable 2-cycle, then a 4-cycle, then 8, 16, ... with the bifurcations arriving at an ever-shrinking rate that converges geometrically to the universal Feigenbaum constant :math:`\delta \approx 4.669`, accumulating at :math:`r \approx 3.5699` onto the onset of chaos. Beyond that value the map is chaotic for most (but not all -- note the periodic windows, the largest starting near :math:`r \approx 3.8284`) values of :math:`r` up to :math:`r=4`; the example below uses :math:`r=3.9`, deep in the chaotic regime. .. GENERATED FROM PYTHON SOURCE LINES 22-32 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.chaos.systems.maps import LogisticMap from physicskit.chaos.visualizers.bifurcation import map_bifurcation_sampler, plot_bifurcation_diagram from physicskit.chaos.visualizers.dynamic_plots import animate_map_cobweb system = LogisticMap(r=3.9) .. GENERATED FROM PYTHON SOURCE LINES 33-40 Animation: the cobweb diagram ------------------------------ The classic way to see a 1D map's dynamics geometrically: bounce between the map curve ``f(x)`` (vertical jumps, computing the next value) and the diagonal ``y = x`` (horizontal jumps, feeding that value back in). For ``r=3.9`` the orbit never settles down -- the cobweb keeps exploring new territory instead of converging onto a fixed point or a short cycle. .. GENERATED FROM PYTHON SOURCE LINES 40-44 .. code-block:: Python anim = animate_map_cobweb(system, x0=0.4, n_iter=40) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 45-49 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("logistic_map_animation.gif", writer="pillow", fps=5) .. GENERATED FROM PYTHON SOURCE LINES 51-53 A single chaotic trajectory ------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 53-61 .. code-block:: Python traj = system.trajectory(np.array([0.4]), n_iter=200) fig, ax = plt.subplots(figsize=(8, 4)) ax.plot(traj[:, 0], lw=1.0, marker=".", markersize=3) ax.set_xlabel("iteration n") ax.set_ylabel("x") ax.set_title(f"Logistic map trajectory, r = {system.r}") .. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_logistic_map_002.png :alt: Logistic map trajectory, r = 3.9 :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_logistic_map_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Logistic map trajectory, r = 3.9') .. GENERATED FROM PYTHON SOURCE LINES 62-69 The bifurcation diagram -------------------------- :func:`physicskit.chaos.visualizers.bifurcation.plot_bifurcation_diagram` sweeps ``r``, iterating each map instance past its transient and plotting the surviving long-term values: a single point for a fixed point, a handful of points for a period-``k`` cycle, and a dense band for chaos. This is the single image most associated with "the route to chaos". .. GENERATED FROM PYTHON SOURCE LINES 69-79 .. code-block:: Python r_values = np.linspace(2.4, 4.0, 2000) sampler = map_bifurcation_sampler(lambda r: LogisticMap(r=r), state0=np.array([0.5]), n_transient=500, n_keep=200) fig2, ax2 = plot_bifurcation_diagram(r_values, sampler) ax2.set_xlabel("r") ax2.set_ylabel("x (long-term values)") ax2.set_title("Logistic map bifurcation diagram") ax2.set_ylim(0.0, 1.0) plt.show() .. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_logistic_map_003.png :alt: Logistic map bifurcation diagram :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_logistic_map_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.784 seconds) .. _sphx_glr_download_api_gallery_chaos_maps_plot_logistic_map.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_logistic_map.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_logistic_map.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_logistic_map.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_