.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/maps/plot_standard_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_standard_map.py: Standard Map ============ The Chirikov-Taylor standard map is an area-preserving (symplectic) map on the cylinder :math:`(\theta, p) \in [0, 2\pi) \times [0, 2\pi)`, obtained by kicking a free rotor once per period with a torque :math:`k \sin\theta`: .. math:: p_{n+1} = p_n + k \sin\theta_n \pmod{2\pi}, \qquad \theta_{n+1} = \theta_n + p_{n+1} \pmod{2\pi} It is the textbook example of the transition from regularity to chaos: at low kick strength :math:`k`, most of phase space is filled with regular invariant curves ("islands"), i.e. :math:`k=0` is integrable since every orbit then lies on a curve of constant :math:`p`; as :math:`k` grows, these islands are progressively destroyed and replaced by a connected chaotic sea, with global chaos setting in around :math:`k \gtrsim 4\text{-}5`. This example reproduces the classic picture by iterating many initial conditions and overlaying their orbits. .. GENERATED FROM PYTHON SOURCE LINES 23-31 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.chaos.systems.maps import StandardMap from physicskit.chaos.utils.metrics import map_lyapunov_spectrum from physicskit.chaos.visualizers.dynamic_plots import animate_multi_orbit_map .. GENERATED FROM PYTHON SOURCE LINES 32-36 Sweep initial momenta ---------------------- For a fixed kick strength, iterate a grid of initial conditions spanning the momentum axis to get one orbit per initial condition. .. GENERATED FROM PYTHON SOURCE LINES 36-41 .. code-block:: Python k = 1.2 system = StandardMap(k=k) initial_momenta = np.linspace(0.05, np.pi - 0.05, 40) orbits = [system.trajectory(np.array([0.0, p0]), n_iter=400) for p0 in initial_momenta] .. GENERATED FROM PYTHON SOURCE LINES 42-49 Animation: the portrait filling in, iteration by iteration ------------------------------------------------------------------- All 40 orbits grow together, one new iterate per frame: watch some initial momenta trace out a closed, quasi-periodic loop around a regular island, while others scatter densely across the connected chaotic sea -- a distinction that is easy to miss once every orbit is fully overlaid at once. .. GENERATED FROM PYTHON SOURCE LINES 49-59 .. code-block:: Python anim = animate_multi_orbit_map( orbits, xlim=(0.0, 2.0 * np.pi), ylim=(0.0, 2.0 * np.pi), labels=(r"$\theta$", "$p$"), title=f"Standard map, k = {k}", ) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 60-64 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("standard_map_animation.gif", writer="pillow", fps=16) .. GENERATED FROM PYTHON SOURCE LINES 66-68 The finished portrait -------------------------- .. GENERATED FROM PYTHON SOURCE LINES 68-80 .. code-block:: Python fig, ax = plt.subplots(figsize=(7, 6)) for traj in orbits: ax.plot(traj[:, 0], traj[:, 1], ",", color="black", alpha=0.6) ax.set_xlim(0.0, 2.0 * np.pi) ax.set_ylim(0.0, 2.0 * np.pi) ax.set_xlabel(r"$\theta$") ax.set_ylabel(r"$p$") ax.set_title(f"Standard map, k = {k}") plt.show() .. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_standard_map_002.png :alt: Standard map, k = 1.2 :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_standard_map_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 81-91 Quantifying the transition: the largest Lyapunov exponent vs. k ----------------------------------------------------------------------- The portrait above only shows the mix of islands and chaotic sea at one kick strength. Sweeping ``k`` and estimating the largest Lyapunov exponent at each value with :func:`~physicskit.chaos.utils.metrics.map_lyapunov_spectrum` (started from a generic initial condition, off any obvious island) turns "the sea grows as k increases" into a number: near-zero for the near-integrable small-k map, then trending positive and increasingly noisy (as the orbit samples islands of different sizes at different k) once global chaos sets in beyond ``k ~= 4-5``. .. GENERATED FROM PYTHON SOURCE LINES 91-105 .. code-block:: Python k_values = np.linspace(0.1, 8.0, 40) lambda_max = np.array([map_lyapunov_spectrum(StandardMap(k=float(k_i)), np.array([0.37, 1.83]), n_iter=4000, n_transient=500)[0] for k_i in k_values]) fig2, ax2 = plt.subplots(figsize=(8, 4.5)) ax2.axhline(0.0, color="black", lw=0.8) ax2.plot(k_values, lambda_max, "o-", color="crimson", markersize=3) ax2.axvline(k, color="gray", ls="--", lw=1.0, label=f"k = {k} (shown above)") ax2.set_xlabel("kick strength k") ax2.set_ylabel(r"largest Lyapunov exponent $\lambda_{max}$ (per iteration)") ax2.set_title("Standard map: chaos strength grows (noisily) with k") ax2.legend(fontsize=8) fig2.tight_layout() plt.show() .. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_standard_map_003.png :alt: Standard map: chaos strength grows (noisily) with k :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_standard_map_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 6.923 seconds) .. _sphx_glr_download_api_gallery_chaos_maps_plot_standard_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_standard_map.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_standard_map.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_standard_map.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_