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

.. only:: html

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

        Click :ref:`here <sphx_glr_download_matplotlib_plot_suris_integrablemap.py>`
        to download the full example code

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

.. _sphx_glr_matplotlib_plot_suris_integrablemap.py:


Suris's integrable map
===================================

.. math::

    p_{n+1} & = p_{n} + \frac{\epsilon}{2\pi}\frac{ \sin2\pi q}{1+\epsilon \cos2\pi q} \\
    q_{n+1} &= q_{n} + p_{n+1}

| [1] Y. B. Suris, Integrable Mappings of the Standard Type, Funct Anal Its Appl 23, 74 (1989).
| [2]J. D. Meiss, Symplectic Maps, Variational Principles, and Transport, Reviews of Modern Physics 64, 795 (1992).
| [3] H. E. Lomelı́ and J. D. Meiss, Heteroclinic Orbits and Flux in a Perturbed Integrable Suris Map, Physics Letters A 269, 309 (2000). 

.. GENERATED FROM PYTHON SOURCE LINES 18-111



.. image:: /matplotlib/images/sphx_glr_plot_suris_integrablemap_001.png
    :alt: $k=0.5$
    :class: sphx-glr-single-img





.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt

    twopi = 2*np.pi
    omega=twopi

    def Map(q,p,eps):
        pp = p + V(q,eps)
        qq = q + pp
        return qq, pp

    def V(q,eps):
        return -np.arctan(
                eps*np.sin(twopi*q) / (1+eps*np.cos(twopi*q) )
            )*2/twopi


    def Phi0(x,y):
        return (1-np.cos(omega*(x-y)))#/omega**2
    def Phi1(x,y):
        return (np.cos(omega*x) + np.cos(omega*y))#/(2*omega)

    def Phi(x,y,eps):
        return np.cos(omega*y) + eps*(np.cos(omega*x) + np.cos(omega*(x-y)))



    def GetTraj(inits, k,sample=100,tmax=500):

        res = [[],[]]
        for q,p in zip(inits[0],inits[1]):
            traj = [np.array([])]*2
            for i in range(tmax):
                traj[0]  = np.append(traj[0], q)
                traj[1]  = np.append(traj[1], p)
                q,p = Map(q,p,eps)
                q = q - np.floor(q)
    #            p = p - np.floor(p)
            res[0].append(traj[0])
            res[1].append(traj[1])
        return res


    import sys
    eps = 0.5

    #tau = 1
    sample=20
    tmax = 200
    x = np.linspace(0,1,1000,endpoint=False)
    y = np.linspace(-1,1,1000)
    X,Y = np.meshgrid(x,y)

    fig1 = plt.figure(figsize=(6,6))
    ax1 = fig1.add_subplot(1,1,1)

    init = [np.array([1/2]),np.array([0])]
    traj = GetTraj(init, eps,sample=sample,tmax=tmax)
    energy = Phi(init[0][0], init[1][0],eps)

    ax1.contour(X,Y,Phi(X,Y,eps),[energy],colors=["k"],linewidths=1.5)

    q0 = 0.5+np.arccos(eps/2)/twopi
    q1 = 1
    p0 = np.arccos(eps**2/2 - 1)/twopi
    p1 = 0

    q = np.linspace(q0, q1, sample)
    p = np.linspace(p0, p1, sample)

    init = [q,p]
    traj = GetTraj(init, eps,sample=sample,tmax=tmax)

    energys = []
    colors = []
    for i in range(sample):
        card = i/sample
        lx, = ax1.plot(traj[0][i],traj[1][i],'o',markersize=4,mec="none", color=plt.cm.Oranges_r(card) )
        energy = Phi(init[0][i], init[1][i],eps)
        energys.append(energy)
        colors.append(lx.get_color())
    clbar = ax1.contour(X,Y,Phi(X,Y,eps),energys, colors=colors)

    ax1.set_ylim(-0.7,0.7)
    ax1.set_xlim(0,1)
    ax1.set_title(r"$k=%.1f$" % (eps),fontsize=16,position=(0.18,1))
    ax1.set_xlabel(r"$q$",fontsize=25)
    ax1.set_ylabel(r"$p$",fontsize=25)

    fig1.tight_layout()
    #fig1.savefig("SurisMap_k%.3f.png" % (k),transparent=True)
    plt.show()


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

   **Total running time of the script:** ( 0 minutes  0.358 seconds)


.. _sphx_glr_download_matplotlib_plot_suris_integrablemap.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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