.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/synchronization/plot_01_kuramoto.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_ode_dynamics_synchronization_plot_01_kuramoto.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_ode_dynamics_synchronization_plot_01_kuramoto.py:


The Kuramoto synchronization transition
=============================================

A thousand oscillators with Lorentzian-distributed natural frequencies
stay incoherent for weak coupling. Above the critical coupling
:math:`K_c = 2\gamma` a synchronized cluster forms, and the steady
coherence follows Kuramoto's exact result :math:`r = \sqrt{1 - K_c/K}`.

.. GENERATED FROM PYTHON SOURCE LINES 12-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.synchronization import KuramotoModel, kuramoto_lorentzian_order_parameter, kuramoto_order_parameter

    n, gamma = 1000, 0.5
    omegas = gamma * np.tan(np.pi * (np.arange(n) + 0.5) / n - np.pi / 2)
    theta0 = np.random.default_rng(0).uniform(0.0, 2.0 * np.pi, n)








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Coherence versus coupling strength
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-41

.. code-block:: Python


    Ks = np.linspace(0.25, 4.0, 16)
    r_sim = []
    for K in Ks:
        traj = KuramotoModel(theta0, omegas, K=K).integrate((0.0, 60.0), dt=0.02, method="rk4")
        r_sim.append(kuramoto_order_parameter(traj.y[-500:])[0].mean())
    K_fine = np.linspace(0.25, 4.0, 300)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(K_fine, kuramoto_lorentzian_order_parameter(K_fine, gamma), "k", label="theory sqrt(1 - Kc/K)")
    axes[0].plot(Ks, r_sim, "o", label=f"simulation, N={n}")
    axes[0].axvline(2 * gamma, color="gray", ls="--")
    axes[0].set_xlabel("coupling K")
    axes[0].set_ylabel("order parameter r")
    axes[0].set_title("Onset of synchrony at Kc = 2 gamma")
    axes[0].legend()




.. image-sg:: /api/gallery/ode_dynamics/synchronization/images/sphx_glr_plot_01_kuramoto_001.png
   :alt: Onset of synchrony at Kc = 2 gamma
   :srcset: /api/gallery/ode_dynamics/synchronization/images/sphx_glr_plot_01_kuramoto_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1192fa510>



.. GENERATED FROM PYTHON SOURCE LINES 42-44

Growth of coherence over time
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-55

.. code-block:: Python


    for K in (0.5, 1.5, 3.0):
        traj = KuramotoModel(theta0, omegas, K=K).integrate((0.0, 30.0), dt=0.02, method="rk4")
        axes[1].plot(traj.t, kuramoto_order_parameter(traj.y)[0], label=f"K={K}")
    axes[1].set_xlabel("t")
    axes[1].set_ylabel("r(t)")
    axes[1].set_title("Coherence r(t)")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 6.937 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_synchronization_plot_01_kuramoto.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_kuramoto.ipynb <plot_01_kuramoto.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_kuramoto.py <plot_01_kuramoto.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_kuramoto.zip <plot_01_kuramoto.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_