{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Complexified KAM curve (standard map)\n\n$k=0.9, \\nu=0.618034$\n\n[1]J. M. Greene and I. C. Percival, Hamiltonian Maps in the Complex Plane, Physica D: Nonlinear Phenomena 3, 530 (1981).\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nimport sys\nimport glob\n\n\nk = -0.9\ntwopi=2.0*np.pi\n\ndef Q(coeff, theta):\n    res = np.copy(theta) + 0.j\n    for i, m in enumerate(coeff[0]):\n        res += -1.j*coeff[1][i]*np.exp(1.j*theta*m) + 1.j*coeff[1][i]*np.exp(-1.j*theta*m)\n    return res\n\ndef P(coeff, theta):\n    return Q(coeff, theta) - Q(coeff, theta + twopi*nu) - np.sin(Q(coeff, theta))*k/2\n\n\n\ndirname = \"nu_0.618034\"\nnu = float(dirname.split(\"_\")[1])\npath = dirname + \"/fourier_coeff_j21.dat\"\ncoeff = np.loadtxt(path).transpose()\n\nfig = plt.figure()\nax = fig.add_subplot(1,1,1,projection='3d')\nax.axis('off')\n\ntraj = np.loadtxt(dirname + \"/traj.DAT\").T\nymin,ymax = 3.5, 4.5\nindex = (traj[1]>ymin) & (traj[1]<ymax)\nindex2 = (traj[1]<3.8) & (traj[0]<4)\ntraj[0][index2] = np.nan\ntraj[1][index2] = np.nan\nindex2 = (traj[1]<4.1) & (traj[0]<0.8)\ntraj[0][index2] = np.nan\ntraj[1][index2] = np.nan\nax.plot(traj[0][index],traj[1][index],',',c='k')\n\nsample=100\nim_max= 0.0\ntheta = np.linspace(0,twopi,sample) + 1.j*im_max\n\nq = Q(coeff, theta)\np = P(coeff, theta)\nax.plot(q.real, -p.real, q.imag,'-k',lw=2)\n\n\nsample=3000\nim_max= 0.083717354931950\ntheta = np.linspace(0,twopi,sample) + 1.j*im_max\n\nq = Q(coeff, theta)\np = P(coeff, theta)\nax.plot(q.real, -p.real, q.imag,'-k',lw=2)\n\nsample=200\n#im_max = 0.0155717354931950\n[re_theta, im_theta] = np.meshgrid(\n    np.linspace(0,twopi, sample),\n    np.linspace(0, im_max, sample)\n)\ntheta = re_theta + 1.j*im_theta\n\nq = Q(coeff, theta)\np = P(coeff, theta)\n#ax.plot_wireframe(q.real, -p.real, q.imag,rstride=20, cstride=1, alpha=0.1,color='k')\nax.plot_wireframe(q.real, -p.real, q.imag,rstride=20, cstride=2, color='k',alpha=0.5)\n\n\nax.set_ylim(ymin,ymax)\nax.set_xlim(0,2*np.pi)\nax.view_init(40,40)\n#plt.savefig(\"KAM.png\",transparent=True)\nplt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## A table of fourier coefficients\n\n\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "print(coeff) # fourier_coeff_j21.dat\""
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.9.7"
    }
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  "nbformat": 4,
  "nbformat_minor": 0
}