{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# 8.5 Numerical calculation of the invariant $c_S$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "This SageMath notebook uses the optional TIDES package which provides an arbitrary precision ode solver.\n",
    "\n",
    "To add the TIDES package to SageMath (on a unix machine) run in a shell  \n",
    "    $ sage -i tides  \n",
    "as instructed at \n",
    "http://doc.sagemath.org/html/en/reference/misc/sage/misc/package.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Definitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "decimal precision = 120 binary precision = 399\n"
     ]
    }
   ],
   "source": [
    "decimal_precision = 120        # number of decimal digits in floating point numbers\n",
    "#\n",
    "binary_precision = ceil(decimal_precision*log(10,2))\n",
    "R = RealField(binary_precision); RealNumber = R\n",
    "print \"decimal precision =\", decimal_precision, \"binary precision =\",binary_precision\n",
    "#\n",
    "solver_step_size = 0.001   # step_size in T for the trajectories returned by the ode solver\n",
    "solver_tolrel = 1e-80      # the relative tolerance for the solver\n",
    "solver_tolabs = 1e-80      # the absolute tolerance for the solver\n",
    "#solver_tolrel = 1e-60      # the relative tolerance for the solver\n",
    "#solver_tolabs = 1e-60      # the absolute tolerance for the solver\n",
    "#\n",
    "sqrt3=R(sqrt(3))\n",
    "nu = (sqrt3-1)/2\n",
    "lambdaplus = 2+sqrt3\n",
    "lambdaminus = 2-sqrt3\n",
    "betaplus=1+sqrt3/3\n",
    "betaminus=1-sqrt3/3\n",
    "#\n",
    "ode_function(T,hminus,hplus)=[-1-hminus^2+lambdaminus*(hplus*hminus+1),\\\n",
    "                              -1-hplus^2+lambdaplus*(hplus*hminus+1)]\n",
    "def calc_traj(point_i,T_f):\n",
    "    [T_i,hminus_i,hplus_i] = point_i\n",
    "    # integrate backwards or forwards?\n",
    "    if T_f < T_i:\n",
    "        signed_solver_step_size = -solver_step_size\n",
    "    else:\n",
    "        signed_solver_step_size = solver_step_size\n",
    "    traj=desolve_tides_mpfr(\n",
    "            ode_function, [hminus_i,hplus_i], T_i, T_f, \n",
    "            signed_solver_step_size,\n",
    "            solver_tolrel, solver_tolabs, decimal_precision)\n",
    "    # prune runaway points\n",
    "    traj_pruned = [point for point in traj if point[1] != NaN and point[2] != NaN]\n",
    "    hh_list = [point[1:] for point in traj_pruned]\n",
    "    return(traj_pruned,hh_list,len(traj)-len(traj_pruned))\n",
    "#\n",
    "#   large T expansion\n",
    "#\n",
    "Ry.<y> = R[]\n",
    "f_poly = 0.0*y\n",
    "v_poly = 0.0*y\n",
    "f_tilde_coeffs = [1.0]\n",
    "v_tilde_coeffs = []\n",
    "def calc_fv_polys():\n",
    "    global f_poly, v_poly\n",
    "    f_tilde_coeffs = [1.0]\n",
    "    v_tilde_coeffs = []\n",
    "    for m_tilde in range(N_tilde_coeffs+1):\n",
    "        ff_sum=sum(f_tilde_coeffs[mp]*f_tilde_coeffs[m_tilde-mp]/binomial(m_tilde,mp) for mp in range(m_tilde+1))\n",
    "        v_tilde_coeffs.append(-2*f_tilde_coeffs[m_tilde] + ff_sum/(2*m_tilde+1))\n",
    "        fv_sum=sum(f_tilde_coeffs[mp]*v_tilde_coeffs[m_tilde-mp]/binomial(m_tilde,mp) for mp in range(m_tilde+1))\n",
    "        f_tilde_coeffs.append(-(m_tilde+1/2)*f_tilde_coeffs[m_tilde]/(m_tilde+1)+(ff_sum+fv_sum)/(m_tilde+1))\n",
    "    f_poly = 0.0*y\n",
    "    v_poly = 0.0*y\n",
    "    for mm in range(N_tilde_coeffs):\n",
    "        f_poly += f_tilde_coeffs[mm]*factorial(mm) *y^(2*mm+1)\n",
    "        v_poly += v_tilde_coeffs[mm]*factorial(mm) *y^(2*mm+1)\n",
    "#\n",
    "#    find point on S using large T expansion\n",
    "#\n",
    "def s_point(T_val):\n",
    "    y_val = 1.0/T_val\n",
    "    f_T_val = f_poly(y=y_val)\n",
    "    v_T_val = -T_val + v_poly(y=y_val)\n",
    "    hminus_val = f_T_val + betaminus * v_T_val\n",
    "    hplus_val = f_T_val + betaplus * v_T_val\n",
    "    return([T_val,hminus_val,hplus_val])\n",
    "#    return({'f_T':f_T_val,'v_T':v_T_val,'hminus':hminus_val,'hplus':hplus_val})\n",
    "#\n",
    "#   solve ode starting from point on S at large T\n",
    "#\n",
    "def separatrix(T,T_f):\n",
    "    point_i=s_point(T)\n",
    "    (s_traj,s_hh_list,number_pruned) = calc_traj(point_i,T_f)\n",
    "    print number_pruned, \" points pruned\", len(s_traj), \" points\"\n",
    "    return(s_traj,s_hh_list)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Integrate the ode from large $T_i$ to somewhat larger $T_f$ to see the instability of the separatrix $S$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Calculate the large $T$ expansion of $S$ to order 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "N_tilde_coeffs =100\n",
    "calc_fv_polys()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Choose $T_i=20$ (arbitrarily).  The function separatrix($T_i$,$T_f$) uses the large $T$ expansion at $T=T_i$ to calculate the initial condition, then integrates the ode to $T_f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 5251  points\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 3,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# T_i = 20, T_f = 25.4\n",
    "(sep_traj,sep_hh_list)=separatrix(20,25.25)\n",
    "list_plot(sep_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Points are pruned from the trajectory when the ode runs away.  The value of $T_f$ is chosen by trial and error to capture the instability."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### try higher order 120 in the large $T$ expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "N_tilde_coeffs =120\n",
    "calc_fv_polys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 5751  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 5,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj,sep_hh_list)=separatrix(20,25.75)\n",
    "list_plot(sep_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "The instability shows up very slightly later in $T$, suggesting that the higher order expansion might be slightly more accurate at placing the initial point on the separatrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Now integrate backwards in $T$ to find the separatrix at large $h_{-}$.  Choose the earliest $T_f$ such that the solver does not run away (i.e., no points are pruned)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1  points pruned 20622  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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OHz6suXPnau3aterRo8dFf48x9NfS0qKRI0fqueeek9T65u9PP/1UL730ku666y5fuY7GgHH8q7feektvvvmm/vM//1NDhw5VdXW15s2bp8zMTN19992+coxj54RrvKza+SloampSYWGhWlpatHjxYr/PutM4MsPURT3yyCNauXKl1q9fr759+/rOp6en6+zZs2poaPArf/ToUV+CT09Pb5fcGxoa1NTUdMEyR48eVXR0tJKTk0NxSWGzfft2HT16VCNGjFB0dLSio6O1ceNGvfDCC4qOjlZaWhpjeBEyMjKUnZ3td+6aa67RoUOHJLVev9R+9uLccfz666/b1X3s2DHLcZTazyJ0Rf/wD/+gBQsWqLCwUMOGDVNRUZEee+wx3+wn49g54Ryv85X5KY1nU1OTCgoKdODAAZWVlfm9VLe7jSOBqYsxxmjOnDlavny5PvjgAw0cONDv8xEjRigmJkZlZWW+c3V1ddq5c6fGjRsnSRo7dqx27typuro6X5m1a9fK6XRqxIgRvjI/rqOtzMiRIxUTExOqywuLG2+8UTU1NaqurvYdI0eO1OzZs32/ZgytjR8/vt2WFnv37lVWVpYkaeDAgUpPT/cbg7Nnz2rjxo1+4+jxeFRZWekrs23bNnk8Hr8ymzZt8tuyYe3atcrMzNSAAQNCdXlhc/r0aV12mf9fxVFRUb5tBRjHzgnneJ3vz3hbHV1dW1jat2+f1q1b1+5/9LrdOIZ1ifkl4Ck5fw8++KBxu91mw4YNpq6uznecPn3aV+aBBx4wffv2NevWrTNVVVXmZz/7WYePxN94442mqqrKrFu3zvTt27fDR+Ife+wxs2vXLvP666//pB6JP9ePn5IzhjG8GJWVlSY6Oto8++yzZt++faa0tNTExcWZN99801empKTEuN1us3z5clNTU2NmzpzZ4ePdw4cPNxUVFaaiosIMGzbM77Hk48ePm7S0NDNz5kxTU1Njli9fbhITE7vk4/Adufvuu02fPn182wosX77cpKSkmCeeeMJXhnH0d+LECbNjxw6zY8cOI8n867/+q9mxY4fv6a1wjdf//u//mqioKFNSUmJ2795tSkpKbPk4/PlcaBybmppMfn6+6du3r6murvb798br9frq6E7jaOv48eKLLxrpGiNdRWD6f63j0P5YsmSJr8z3339v5syZY3r37m169uxpbrnlFnPo0CG/er788kszffp007NnT9O7d28zZ84cv8c+jTFmw4YNJjc318TGxpoBAwaYl156KRyXGBHnBibG8OKsWrXK5OTkGKfTaYYMGWJeffVVv89bWlrMb3/7W5Oenm6cTqf5m7/5G1NTU+NX5ttvvzWzZ882LpfLuFwuM3v2bNPQ0OBX5pNPPjE33HCDcTqdJj093Tz99NNd+lH4H2tsbDRz5841/fv3Nz169DCDBg0yTz75pN8/Soyjv/Xr13f49+Ddd99tjAnveL399tvm6quvNjExMWbIkCFm2bJlIb32YLrQOB44cOC8/96sX7/eV0d3GkeHMfbe5rV1zVejJLckj6RE2bvHAADgp4Y1TAAAABYITAAAABYITAAAABYITAAAABYITAAAABYITAAAABYITAAAABYITAAAABZsHZgWLVokKVvSqEh3BQAAdGPs9A0AAGDB1jNMAAAAdkBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsGDrwMRO3wAAwA7Y6RsAAMCCrWeYAAAA7IDABAAAYIHABAAAYIHABAAAYCEsgemee+6Rw+HwO8aMGROOpgEAAAIWHa6Gpk2bpiVLlvh+jo2NDVfTAAAAAQlbYHI6nUpPTw9XcwAAAEETtjVMGzZsUGpqqq666irdd999Onr06HnLer1eNTY2qrGxUa17MDWGq5sAAADthGXjyrfeeksJCQnKysrSgQMH9NRTT6m5uVnbt2+X0+lsV/7pp5/WM88800FNbFwJAADCL+iBqbS0VPfff7/v5/fee0833HCDX5m6ujplZWVp6dKl+sUvftGuDq/XK6/XK0lyu6XWGaZ+IjABAIBICPoapvz8fOXl5fl+7tOnT7syGRkZysrK0r59+zqsw+l0djjzBAAAEAlBD0wul0sul+uCZb799lsdPnxYGRkZwW4eAAAg6EK+6PvkyZN6/PHHVVFRoYMHD2rDhg269dZblZKSor/7u78LdfMAAAABC/m2AlFRUaqpqdGf/vQnHT9+XBkZGZo8ebLeeusty5koAAAAOwjLU3KBcDik1kXfbrHoGwAARALvkgMAALBAYAIAALBAYAIAALBg68C0aNEiSdmSRkW6KwAAoBtj0TcAAIAFW88wAQAA2AGBCQAAwAKBCQAAwAKBCQAAwAKBCQAAwAKBCQAAwAKBCQAAwAKBCQAAwIKtAxM7fQMAADtgp28AAAALtp5hAgAAsAMCEwAAgAUCEwAAgAUCEwAAgAUCEwAAgAUCEwAAgAUCEwAAgAUCEwAAgAVbByZ2+gYAAHbATt8AAAAWbD3DBAAAYAcEJgAAAAsEJgAAAAsEJgAAAAsEJgAAAAsEJgAAAAsEJgAAAAsEJgAAAAu2Dkzs9A0AAOyAnb4BAAAs2HqGCQAAwA4ITAAAABYITAAAABYCDkzLly/X1KlTlZKSIofDoerq6nZlvF6vHnnkEaWkpCg+Pl75+fn66quvAm0aAAAgLAIOTKdOndL48eNVUlJy3jLz5s3Tu+++q6VLl2rLli06efKkbrnlFv3www+BNg8AABByQXtK7uDBgxo4cKB27Nih6667znfe4/Ho8ssv15///Gf98pe/lCTV1taqX79++stf/qKpU6deuIM8JQcAACIs5GuYtm/frqamJk2ZMsV3LjMzUzk5OSovLw918wAAAAGLDnUD9fX1io2NVVJSkt/5tLQ01dfXd/gdr9crr9f7ozONIewhAADAhXVqhqm0tFQJCQm+Y/PmzZfcsDFGjtb7be0UFxfL7XbL7Xar9VZcv0tuBwAAIFCdCkz5+fmqrq72HSNHjrT8Tnp6us6ePauGhga/80ePHlVaWlqH31m4cKE8Ho88Ho9a1y0d7kw3AQAAgqpTgcnlcunKK6/0HT179rT8zogRIxQTE6OysjLfubq6Ou3cuVPjxo3r8DtOp1OJiYlKTEyU1HYAAABERsBrmL777jsdOnRItbW1kqTPPvtMUuvMUnp6utxut371q1/pN7/5jZKTk9W7d289/vjjGjZsmG666aZAmwcAAAi5gJ+SW7lypXJzczV9+nRJUmFhoXJzc/Xyyy/7yvzbv/2bbrvtNhUUFGj8+PGKi4vTqlWrFBUVFWjzAAAAIRe0fZhChX2YAABApPEuOQAAAAsEJgAAAAsEJgAAAAsEJgAAAAu2DkyLFi2SlC1pVKS7AgAAujGekgMAALBg6xkmAAAAOyAwAQAAWCAwAQAAWCAwAQAAWCAwAQAAWCAwAQAAWCAwAQAAWCAwAQAAWLB1YGKnbwAAYAfs9A0AAGDB1jNMAAAAdkBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsEBgAgAAsGDrwMTGlQAAwA7YuBIAAMCCrWeYAAAA7IDABAAAYIHABAAAYIHABAAAYIHABAAAYIHABAAAYIHABAAAYIHABAAAYMHWgYmdvgEAgB2w0zcAAIAFW88wAQAA2AGBCQAAwAKBCQAAwAKBCQAAwELAgWn58uWaOnWqUlJS5HA4VF1d3a7MpEmT5HA4/I7CwsJAmwYAAAiLgAPTqVOnNH78eJWUlFyw3H333ae6ujrf8corrwTaNAAAQFhEB1pBUVGRJOngwYMXLBcXF6f09PRAmwMAAAi7sK1hKi0tVUpKioYOHarHH39cJ06cOG9Zr9erxsZGNTY2qnUPpsZwdRMAAKCdgGeYLsbs2bM1cOBApaena+fOnVq4cKE+/vhjlZWVdVi+uLhYzzzzTDi6BgAAYKlTO32Xlpbq/vvv9/383nvv6YYbbpDUektu4MCB2rFjh6677roL1rN9+3aNHDlS27dv1/XXX9/uc6/XK6/XK0lyu6XWGaZ+YqdvAAAQCZ2aYcrPz1deXp7v5z59+lxSo9dff71iYmK0b9++DgOT0+mU0+m8pLoBAACCrVOByeVyyeVyBdzop59+qqamJmVkZARcFwAAQKgFvIbpu+++06FDh1RbWytJ+uyzzyRJ6enpSk9P1xdffKHS0lLdfPPNSklJ0a5du/Sb3/xGubm5Gj9+fKDNAwAAhFzAT8mtXLlSubm5mj59uiSpsLBQubm5evnllyVJsbGxev/99zV16lRdffXVevTRRzVlyhStW7dOUVFRgTYPAAAQcp1a9B0JDofUuujbLRZ9AwCASOBdcgAAABYITAAAABYITAAAABZsHZief36RpGxJoyLdFQAA0I3ZetF3dbWUmyux6BsAAESSrWeYAAAA7MDWgamlJdI9AAAAsHlgamqKdA8AAABsHph++CHSPQAAALB5YAIAALADWwcmZpgAAIAd2DowsegbAADYga0DEwAAgB3YOjAtXcpO3wAAIPJsHZhych6WtEvSh5HuCgAA6MZsHZi+/z7SPQAAALB5YOKdcQAAwA5sHZjOno10DwAAAGwemLglBwAA7MDWgQkAAMAObB2YTp+OdA8AAABsHph4NQoAALADWwempqZI9wAAAMDmgWnnTnb6BgAAkWfrwHTllez0DQAAIs/WgenYsUj3AAAAwOaBCQAAwA5sHZh4NQoAALADWwcmrzfSPQAAALB5YGKGCQAA2IGtAxMv3wUAAHZg68DELTkAAGAHtg5M7PQNAADswNaB6ehRdvoGAACRZ+vA1Ls3O30DAIDIs3Vg4pYcAACwA1sHJp6SAwAAdmDrwNTSEukeAAAA2DwwNTdHugcAAAABBqampibNnz9fw4YNU3x8vDIzM3XXXXeptrbWr1xDQ4OKiorkdrvldrtVVFSk48ePW9bPPkwAAMAOAgpMp0+fVlVVlZ566ilVVVVp+fLl2rt3r/Lz8/3KzZo1S9XV1VqzZo3WrFmj6upqFRUVBdRxAACAcHEYE9w3tn344YcaPXq0vvzyS/Xv31+7d+9Wdna2tm7dqry8PEnS1q1bNXbsWO3Zs0dXX331eeu6/HLpm28kqVGSW5JHUiLvmAMAAGEV9DVMHo9HDodDvXr1kiRVVFTI7Xb7wpIkjRkzRm63W+Xl5R3W4fV61djYqO+/b1RrWGoMdjcBAAAuWlAD05kzZ7RgwQLNmjVLiYmJkqT6+nqlpqa2K5uamqr6+voO6ykuLpbb7dapU261ziz1C2Y3AQAAOqVTgam0tFQJCQm+Y/Pmzb7PmpqaVFhYqJaWFi1evNjvew6Ho11dxpgOz0vSwoUL5fF4FBvrUettuMOd6SYAAEBQRXemcH5+vt+ttT59+khqDUsFBQU6cOCAPvjgA9/skiSlp6fr66+/blfXsWPHlJaW1mE7TqdTTqezM10DAAAImU4FJpfLJZfL5XeuLSzt27dP69evV3Jyst/nY8eOlcfjUWVlpUaPHi1J2rZtmzwej8aNG3fB9ng1CgAAsIOAnpJrbm7W7bffrqqqKq1evdpvxqh3796KjY2VJP385z9XbW2tXnnlFUnSr3/9a2VlZWnVqlUX7pzvjh1PyQEAgMgJKDAdPHhQAwcO7PCz9evXa9KkSZKk7777To8++qhWrlwpqfXW3osvvuh7ku68nSMwAQAAGwj6PkzBRGACAAB2YOt3yQEAANgBgQkAAMACgQkAAMCCzQPTIknZkkZFuiMAAKAbY9E3AACABZvPMAEAAEQegQkAAMACgQkAAMACgQkAAMACgQkAAMACgQkAAMACgQkAAMACgQkAAMCCzQMTO30DAIDIY6dvAAAACzafYQIAAIg8AhMAAIAFAhMAAIAFAhMAAIAFAhMAAIAFAhMAAIAFAhMAAIAFAhMAAIAFmwcmdvoGAACRx07fAAAAFmw+wwQAABB5BCYAAAALBCYAAAALBCYAAAALBCYAAAALBCYAAAALBCYAAAALBCYAAAALNg9M7PQNAAAij52+AQAALNh8hgkAACDyCEwAAAAWCEwAAAAWCEwAAAAWAgpMTU1Nmj9/voa51EX9AAAG00lEQVQNG6b4+HhlZmbqrrvuUm1trV+5AQMGyOFw+B0LFiwIqOMAAADhEtBTch6PRzNmzNB9992na6+9Vg0NDZo3b56am5v10Ucf+coNGDBAv/rVr3Tffff5ziUkJCghIeHCneMpOQAAYAPRgXzZ7XarrKzM79y///u/a/To0Tp06JD69+/vO+9yuZSenh5IcwAAABER9DVMHo9HDodDvXr18jv/hz/8QcnJybruuuv07LPP6uzZs+etw+v1qrGxUa0zS20HAABAZAQ0w3SuM2fOaMGCBZo1a5YSExN95+fOnavrr79eSUlJqqys1MKFC3XgwAH9x3/8R4f1FBcX65lnnglm1wAAAC5Zp9YwlZaW6v777/f9/N577+mGG26Q1LoA/I477tChQ4e0YcMGv8B0rmXLlmnGjBn65ptvlJyc3O5zr9crr9crt7vtTKOkfmINEwAAiIROBaYTJ07o66+/9v3cp08f9ezZU01NTSooKND+/fv1wQcfdBiCfuzIkSPq27evtm7dqry8vPN3jkXfAADABjp1S87lcsnlcvmdawtL+/bt0/r16y3DkiTt2LFDkpSRkdGZ5gEAACIioDVMzc3NmjFjhqqqqrR69Wr98MMPqq+vlyT17t1bsbGxqqio0NatWzV58mS53W59+OGHeuyxx5Sfn+/3FB0AAIBdBbQP08GDBzVw4MAOP1u/fr0mTZqkqqoqPfTQQ9qzZ4+8Xq+ysrJUWFioJ554QnFxcRfuHLfkAACADQQUmEKNwAQAAOyAd8kBAABYIDABAABYIDABAABYsHlgWiQpW9KoSHcEAAB0Yyz6BgAAsGDzGSYAAIDIIzABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYsHlgYqdvAAAQeez0DQAAYMHmM0wAAACRR2ACAACwQGACAACwQGACAACwQGACAACwQGACAACwQGACAACwQGACAACwYPPAxE7fAAAg8tjpGwAAwILNZ5gAAAAij8AEAABggcAEAABggcAEAABggcAEAABggcAEAABggcAEAABggcAEAABgweaBiZ2+AQBA5LHTNwAAgAWbzzABAABEHoEJAADAAoEJAADAQsCB6emnn9aQIUMUHx+vpKQk3XTTTdq2bZtfmYaGBhUVFcntdsvtdquoqEjHjx8PtGkAAICwCDgwXXXVVXrxxRdVU1OjLVu2aMCAAZoyZYqOHTvmKzNr1ixVV1drzZo1WrNmjaqrq1VUVBRo0wAAAGER9KfkGhsb5Xa7tW7dOt14443avXu3srOztXXrVuXl5UmStm7dqrFjx2rPnj26+uqrz985npIDAAA2ENQ1TGfPntWrr74qt9uta6+9VpJUUVEht9vtC0uSNGbMGLndbpWXlwezeQAAgJCIDkYlq1evVmFhoU6fPq2MjAyVlZUpJSVFklRfX6/U1NR230lNTVV9fX2H9Xm9Xnm93h+daQxGNwEAAC5Jp2aYSktLlZCQ4Ds2b94sSZo8ebKqq6tVXl6uadOmqaCgQEePHvV9z/HXe2s+xpgOz0tScXGx3G63Wm/DuSX160w3AQAAgqpTgSk/P1/V1dW+Y+TIkZKk+Ph4XXnllRozZoxef/11RUdH6/XXX5ckpaen6+uvv25X17Fjx5SWltZhOwsXLpTH41HrmiWPpMOduigAAIBg6tQtOZfLJZfLZVnOGOO7pTZ27Fh5PB5VVlZq9OjRkqRt27bJ4/Fo3LhxHX7f6XTK6XR2pmsAAAAhE9BTcqdOndKzzz6r/Px8ZWRk6Ntvv9XixYv15ptvavv27Ro6dKgk6ec//7lqa2v1yiuvSJJ+/etfKysrS6tWrbpw53hKDgAA2EBAT8lFRUVpz549uv3223XVVVfplltu0bFjx7R582ZfWJJa1z4NGzZMU6ZM0ZQpUzR8+HD9+c9/DrjzAAAA4RD0fZiCiRkmAABgB7xLDgAAwAKBCQAAwAKBCQAAwILNA9MiSdmSRkW6IwAAoBtj0TcAAIAFm88wAQAARB6BCQAAwAKBCQAAwAKBCQAAwAKBCQAAwIKtA9Pp0xd3DgAAIJRsHZh69pTq6v768969recAAADCydaBSZLS0yWPp/XXaWmR7QsAAOiebB2YFi1apOzsbI0axU7fAAAgcmy903ebxsZGud1ueTweJSYmRro7AACgm7H1DBMAAIAdEJgAAAAsdIlbcsYYnThxQi6XS46/vpEXAAAgLLpEYAIAAIgkbskBAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABYIDABAABY+D+P54qks3sBgQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 6,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj_back,sep_back_hh_list) = separatrix(20,-0.6220)\n",
    "list_plot(sep_back_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 20622  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 7,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj_back,sep_back_hh_list) = separatrix(20,-0.621)\n",
    "list_plot(sep_back_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Get the earliest point on the trajectory."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.62099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998,\n",
       " 11979.738443413583082144782481069482787725314129673935381110229234360188556339628695461441469072904293690294961463237222,\n",
       " -0.000083474276564673047602103434215648075436230636068420002091783368642468601744304745701367323280449836251819145030300315359]"
      ]
     },
     "execution_count": 8,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "back_point = sep_traj_back[-1]; back_point"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Integrate foward from that earliest point until just before runaway."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 9,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj_forward,sep_forward_hh_list,number_pruned) =\\\n",
    "    calc_traj(back_point,12.82)\n",
    "list_plot(sep_forward_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Plot the points on the trajectory with $h_{-} < 10.0$, displaying the instability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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sUV5enkaNGqV//OMfysrKMt0eXOz226VHHjn26zZulAoKYt8PzNi0aZOmTJmiHTt2qHPnzho6dKg+/vhj9ejRw3RriBOfffaZRo0aVTf+xS9+IUmaOnWq/va3v+n222/XwYMHdeONN2r37t0qLi7Wu+++6/qfUZ5AgP8jAonM7/fL6/XK5/MpOzs76PmmzCbl50ubN8egOQBwAWaWgCTWlKDEf6cAJDsWeANJ6lhBKS+PoAQAEjNLQFI6VlAiJAFAPWaWgCSyfTtBCQCai7AEJImyMqlLl8ZfQ1ACgGCEJSBJjB4d/rkJEwhKABAOa5aABHesm8avWiX16eNMLwDgRuyzBCSwmhopJcUvySvJJ8m+zxL/+gHg2DgNBySwlJTwzxGUAKBpCEtAgvJ4SiQVSioKeo6gBABNx2k4IAHZtwewn4bjXzwANA8zS0CCaWwfpQMHnOsDABIFYQlIII0FpYcektq2da4XAEgUhCUgQQwZ0vjzN9zgTB8AkGgIS0CCKCsz3QEAJCbCEpAAGjv95vM51wcAJCLCEuByjQUlrnwDgMgRlgAXIygBQOwRlgCX2r49/HNPP+1cHwCQ6NiUEnCpps4q+f1+eb1e+Xw+ZWdnh/8kAEBIzCwBLsTpNwBwDmEJcJl+/cI/R1ACgOgjLAEus2pV6Pr//Z+zfQBAsiAsAS7S2Om3iROd6wMAkglhCXAJ1ikBgBmEJcDlwgWlkpISFRYWqqioyNmGACDBsHUA4ALhZpVSU6XDhxv/XLYOAIDIMLMExLnGTr8dKygBACJHWAJcijlhAHAGYQmIY+FmlXJynO0DAJIZYQmIUx07hn+usfvCAQCii7AExKndu0PXOf0GAM4iLAFxKNzpt8YWewMAYoOwBMSZhx8O/1xNjXN9AAAs7LMExJlws0ct/ZfKPksAEBlmloA4wmk2AIg/hCXABZj/BQBzCEtAnAg3q/TYY872AQCwIywBcWDUqPDP/b//51wfAIBghCUgDixYELrO6TcAMI+wBBjGom4AiG+EJSBORTqrVFJSosLCQhUVFUWnIQBIUuyzBBgUblZp3jzp7LOjcwz2WQKAyDCzBBgyY0b456IVlAAAkYvZzNKOHdLHH0u9e0snnhiLIwDuFu2dusNhZgkAIhOTmaVvv5VOOkmaMEEaNEh6+eVYHAVwLxZ1A4B7xCQs/e1v0tat1q+rq6XbbpM2bozFkYDEwgpCAIg/MQlLXq99XF4uFRVJW7bE4miAu4SbVbrhBmf7AAA0TUzC0g03SOPG2Wvbtkm33ipVVsbiiIA7HD4c/rk//cm5PgAATReTsNSmjfTmm9LAgfb67NnSNdfE4oiAO6Snh65z+g0A4ldMtw54+WWpVy97bfZsac6cWB4ViE/TppnuAADQEjHflHLRIum00+z/c27VyqoXF8fyyEB8cWqrgIbYOgAAIhPzTSmHD5dmzpTS0uprNTXSjTdK330X66MD8YGtAgDAvRzZwXvKFOn88+21JUukq65y4uhA/GrKrNK0adPk8Xhsj6FDh8a+OQCAJAdvd/K3v0ljxthr8+dLzzzjVAeAGeFmla64ounvce6552rr1q11j7lz50anOQDAMTkWltq2lR57zPpYq7pa+tnPpM8+c6oLIH688ELTX5uenq7c3Ny6R8eOHWPXGADAxtEb6RYWSh99ZC3wrhUISD/9qbR9u5OdAM4IN6u0Z0/z3mfBggXq0qWL+vbtq2uvvVbbtm0L+9rKykr5/X7bAwDQco6GJUk65RTp8svttS++kH7xC6c7AWKroiL8cw13uW/M2LFjNXPmTM2fP1+PPvqoysrKdNZZZ6kyzA6vDzzwgLxeb92joKCgmZ0DAI4W860DQjlyRBo7Vnrvvfpaerr00kvShRc63Q0QGy3ZKmDmzJm6/vrr68ZvvfWWTj/9dNtrtm7dqh49emjWrFm6MMQ/mMrKSluQ8vv9KigoYOsAAGih1iYOmpIi/e530sKF9bc/qayULrvMun8cyzHgdjff3LLPmzhxooqP2oCsW7duQa/Jy8tTjx49tHr16pDvkZ6ervRwW4UDAJrN8dNwtYqLpX/+016rrLTuK3fkiJmegGh56qnQ9WPN42ZlZal37951j7ZHXxHxvZ07d6q8vFx5eXlR6BQAcCzGwpIkjR8vnXGGvfbyy9Kf/2ymHyAaMjOj91779u3TjBkztHjxYm3YsEELFizQhAkTlJOTowsuuCB6BwIAhGU0LKWkSO+8I/Xsaa///vfS+vVmegIitX9/6HpLVgempKToyy+/1Hnnnae+fftq6tSp6tu3rxYvXqysrKzIGgUANImRBd4NPfOMdfrtaKNGWZtWAm4SblH30KHS4sXO9lKLe8MBQGSMzizVuv764LD0739Lb71lph8g2kwFJQBA5OIiLEnSbbfZ9545fFiaNEnascNcT0BzhJtVmjPH2T4AANEVN2Gpd+/gHyoHD0oPPWSmHyBaWIcNAO4WN2FJsq6Ma3gz9f/+b+m118z0AzRVuFmlMJtsAwBcJK7CUuvW0rx5UufO9vrjj1un5QC3SUsz3QEAIFJxFZYkqV07aepUe23BAumee4y0AxxTS25rAgBwj7gLS5L08MPSyJH22j/+IXHzdMSbr7823QEAINbiMix5PNLPf26vffutdNFFZvoBwunfP3SdWSUASBxxGZYka9uAW26x1+bPZysBxI/f/c50BwAAJ8RtWJKsjSqPXiB75Ih0+unWlgKAab/5Teh6vMwqlZSUqLCwUEVFRaZbAQBXi4vbnTTm+eeDF3y//7501llm+gEkacQIadGi0M/F278obncCAJGJ+7C0f79UUCDt3l1fy8+Xli6VunQx1xeSm5uugCMsAUBk4vo0nGRtJfDmm9YeTLW2bJFmzzbXE5JbVlboesP9wQAAiSHuZ5ZqDRworVhRP87JkT77TOrRw1xPSE5umlWSmFkCgEjF/cxSrRdftGaZau3YIZWUmOsHySlcUBo71tk+AADOcU1YOvnk4I0q//IXad06I+0ANnPnmu4AABArrglLkvTQQ1JmZv14925ugwLnhJtVevRRZ/sAADjLVWFpwADp8svttXfekTZuNNMPIEm/+IXpDgAAseSqsCRJt95qvxpp507puuvM9YPkEG5WqazM2T4AAM5zXVjq3z/4NiiffMJtUGDGD39ougMAQKy5LixJ0hVX2Ncu7dkjXXqpuX6Q2MLNKh29USoAIHG5MiydeKL061/ba4sWSZWVZvpBcmrf3nQHAAAnuDIsSdJ550lt2tSPDx2SJk0y1w8Sk9s2oAQARJ9rw9KJJ0r33Wevvfkms0sAACC6XBuWJGn8eCklxV679VYzvSDxMKsEAJBcHpZOPFH6r/+y1555xtpOAEh2JSUlKiwsVFFRkelWAMDVXHMj3XC+/traTuBojzwizZhhph8khkSaVeJGugAQGVfPLEnW7NIvf2mv3XGHtH27mX7gfm4MRACA2HF9WJKkn/zEPq6pkd5/30wvcL9WYf5VEKIAIDklRFjq31+68EJ77brrrM0qgeY4cMB0BwCAeJMQYUmSfv97+3jvXumLL8z0Avdq1y50nVklAEheCROWTjhB+sEP7LVrrrE2qwSaYutW0x0AAOJRwoSl1FTpz3+219aulVavNtMP3Cc/P3SdWSUASG4JE5Yk68q4nBx77Ve/4ocdjm3FCtMdAADiVUKFpawsa1PKo73+urRunZl+4B4DB4auE7QBAAkVliRp2LDgW6DMmWOmF7jD/PmmOwAAxLOEC0t5edK999prd90l7d5tpB24wNlnh64zqwQAkBIwLEnSxIn28ZEjbCOA0F56yXQHAIB45/p7w4USCEjjxklvv11fO+446dtvw+/OjOSUSPeAC4d7wwFAZBIyOng8wTfS3bSJ+8XB7umnTXcAAHCDhAxLknTqqcH75tx0k5leEJ9+9rPQ9USaVQIARC5hw1L79tJvfmOvvfoqPwhhue8+0x3EXklJiQoLC1VUVGS6FQBwtYRcs1Triy+kU06Ramrqay++KF1+ubmeEB+SYa1SLdYsAUBkEnZmSbLuFfeTn9hrd99tphfEj7vuMt0BAMBNEjosSdKZZ9rH27ZJ5eVmekF8ePDB0PVEnFUCAEQu4cPS5MnSgAH140OHpJ//3Fw/MGv69ND1hru+AwBQK+HDUmqqdP759tqyZWZ6gXl/+EPoenW1s30AANwj4cOSJE2ZIrVpUz/euDH8D00krhtvDF1PS3O2DwCAuyRFWBowQPrRj+y1d94x0wvM+dOfQtcrK53tAwDgLkkRliRp9Gj7eMEC6ZtvjLQCA665JnS9bVtn+wAAuE/ShKWbb5Z69KgfHzwoPfecuX7grL/+NXT9wAFn+wAAuE/ShCWPRzrpJHvtvfe4XDwZTJsWup6Z6WgbAACXSugdvBtat07q3186fLi+9uqr0nnnmesJsZdMu3WHwg7eABCZpJlZkqRevaSsLHtt6VIzvcAZV10Vut7w6wAAgHCSKixJ0h132MePPy7t3GmmF8TeCy+Ervv9zvYBAHCvpAtLkyfbx36/tHq1mV4QW+HWKjGrBABojqQLS3l50okn2muPP26mF8RWuKsdk2VWqaSkRIWFhSoqKjLdCgC4WlIt8K71wgv2tSytWlkbE7Zuba4nRNfVV0vPPhtc93qlPXuc78ckFngDQGSSbmZJkgYNsl8hFQhIZWXm+kH0hQpKUvIFJQBA5JIyLJ18sn27gEBAuv12c/0gun72s9B11ioBAFoiKcOSJDVcxrFmjXTkiJleEF1PPx26nixrlQAA0ZW0Yem666QOHerHFRXSk0+a6wfRcfPNoevt2jnbBwAgcSRtWMrJsU7HHY11S+731FOh6/v2OdsHACBxJG1YkoL3XHrtNam83EwviFzDDUdrZWQ42wcAILEk5dYBR0tNlaqr68cvvSRNmWKuH7Rcst8DLhy2DgCAyCT1zJJkbSNwtDffNNMHInPPPaHraWnO9gEASDxJP7P0xhvShAn22o4dUqdOZvpByzCrFB4zSwAQmaSfWerVK7i2caPzfaDlHn44dD0lxdk+AACJKenDUmGhNG6cvcYGle4SbmH30WvR4tWcOXM0ZswY5eTkyOPxaNmyZUGvqays1M0336ycnBy1a9dOEydO1KZNmwx0CwDJKenDkiQNH24ff/21mT7QfM88Y7qDyOzfv18jRozQgw8+GPY106dP17/+9S/NmjVLH330kfbt26cf//jHOsIuqgDgiKRfsyRZu3cPGiQdOlRfmz1bmjTJXE9omkRZq7Rhwwb17NlTS5cu1clHbQDm8/nUuXNnvfDCC7r00kslSVu2bFFBQYHmzp2rMWPGHPO9WbMEAJFhZklS797W6bijlZaa6QVN99JLpjuIvc8//1xVVVU655xz6mr5+fkaOHCgFi1aFPJzKisr5ff7bQ8AQMsRlr531ln28Rtv2GeaEH8uvzx03W2zSo2pqKhQWlqaOhx9bx5JXbt2VUVFRcjPeeCBB+T1euseBQUFTrQKAAmLsPS9O++0j9evlz791EwvOLa5c0130HwzZ85UZmZm3ePDDz9s8XsFAgF5wpyDvOuuu+Tz+eoe5WxLDwARaW26gXiRnS1lZUl799bXVqyQzjjDXE8Ib/z40PV4nlWaOHGiiouL68bdunU75ufk5ubq8OHD2r17t212adu2bRre8MqE76Wnpys9PT3yhgEAkphZqpOaKt16q712331mekHjwizViXtZWVnq3bt33aNt27bH/JxTTz1Vqampeu+99+pqW7du1fLly8OGJQBAdDGzdJSGi7x37ZKqqqwghfgxYkToejzPKoWza9cubdy4UVu2bJEkffPNN5KsGaXc3Fx5vV5dc801uu2229SpUyd17NhRM2bM0KBBgzR69GiTrQNA0mBm6Sjnny/l5dWPDx2SHnnEXD8ItmKF6Q6i67XXXtPgwYM1/vvzipMnT9bgwYP19NNP173m8ccf1/nnn69LLrlEI0aMUEZGhl5//XWlsEU5ADiCfZYa+PGP7TfTvfBC6ZVXzPUDu0TZV8lJ7LMEAJFhZqmB7/f9qzNvnrRtm5leYPf9mSoAABxFWGrgoovsY79fWrLETC+wC3cNWhjEAAASKklEQVTxGLNKAIBYIiw10LatdMIJ9tr8+WZ6Qb3du013AABIVoSlEK6/3j5+9lkzfaBex46h68wqAQBijbAUQr9+9vHevcxsmHTwoOkOAADJjLAUwsSJ0vHH148rK6XHHjPWTtLLyAhdZ1YJAOAEwlIYffrYx4m2v49b1NSY7gAAkOwIS2FMn24fz50rffedmV6SWbh9F5lVAgA4hbAURsNbn1RWShs3mukFAACYQ1gKo6AgODC98IKZXpIVu3VHpqSkRIWFhSoqKjLdCgC4Grc7acTtt9vvDTdwoPTll+b6STaEpejgdicAEBlmlhrR8O72a9dyyw2nEJQAAPGCsNSI886TsrLqxwcP2m+yCwAAEh9h6Rga3vqE+8TFHrNKAIB4Qlg6hp/+1D5+/nl+aAMAkEwIS8fQcGbp8GFp/34zvSQDZpUAAPGGsHQMY8ZIxx1XP66ulh5/3Fw/AADAWYSlY/B4pL597bXly830kuiYVQIAxCPCUhP85Cf28ZtvSvv2mekFAAA4i7DUBAMG2Mf790vbtpnpJVExqwQAiFeEpSYoLJTy8+21f/zDTC8AAMBZhKUmSE+Xhg+31z780EwviYhZJQBAPCMsNdHZZ9vHn3zCuiUAAJIBYamJJk60j3ftklasMNNLIgk3q3TwoLN9AAAQDmGpiXJypI4d7bWyMjO9JIM2bUx3AACAhbDURGlp1gaVR3vlFTO9JIpws0o7djjbR6IqKSlRYWGhioqKTLcCAK7mCQRYRttUjzwi3X57/bhLF2nTJik11VxPbsbCbmf4/X55vV75fD5lZ2ebbgcAXIeZpWa49lr7eNs2adEiM724Xbig9M03zvYBAMCxEJaaoV274LU0a9ea6SVRNby1DAAAphGWmiE1VRo3zl578UUzvbhZuFml995ztg8AAJqCsNRMgwbZx6tWmekjEY0ebboDAACCEZaa6YYb7OPNm6WPPzbTixuFWwz/v//rbB8AADQVYamZ2rcPPo20bp2ZXtyoujp0veHieQAA4gVhqZnatJFGjbLXZs0y04vbdOkSun70dgwAAMQbwlILnHKKfbxypZk+3Gb79tD1hx5ytg8AAJqDsNQCU6fax2vWSF9/baYXt2i4ML7WRRc52wcAAM1FWGqBvLzg2vr1zvfhJsuXh66//LKzfQAA0FyEpRbo1Ek66SR7jR/64f34x6HrP/yhs30AANAShKUWanhv0v/8x0wfbvDmm6HrZWXO9gEAQEsQllpo0iT7eOnS8AuYk9mMGaHrxx3nbB8AALQUYamFeva0j2tqrA0qYffoo6Hr5eXO9gEAQEsRllqob18pP99emz3bTC/xKtyu3BkZzvaRrEpKSlRYWKiihueMAQDN4gkEAgHTTbjV2LHS22/Xj889V3rrLXP9xJtwN8zlK85Zfr9fXq9XPp9P2dnZptsBANdhZikC555rH5eWWqfjIC1caLoDAACig7AUgQED7OODB6V9+8z0Em/OPDN0nVklAIDbEJYicPrpUrt29trf/26ml3jC4m0AQCIhLEUgPT34EvjPPzfTSzzp3j10nVklAIAbEZYi1PB00/z5ZvqIFwcOmO4AAIDoIixF6NRT7eOtW6XqajO9xIOGpyVrMasEAHArwlKELr3UPj5wQPrgAzO9AACA6CMsRSgrS0pLs9eS9T5x7KsEAEhEhKUItWolDRtmr5WWmukFAABEH2EpChquW/rss+SbTWFWCQCQqAhLUTBmjH28dau0bZuZXgAAQHQRlqKgf3/rdNzRli8304sJzCoBABIZYSkKCgqk3Fx77f33zfQCAACii7AUJf362cfJsn0As0oAgERHWIqSUaPs4zVrzPQB1CopKVFhYaGKiopMtwIAruYJBJgDiIbFi6Xhw+21VaukPn3M9OMEZpXcwe/3y+v1yufzKTs723Q7AOA6zCxFSe/ewbUVK5zvAwAARBdhKUo6d5a6dbPXEnndUrhZpcpKZ/sAACDWCEtRNHCgffzhh2b6MKnhrV8AAHA7wlIUnX66fbxqlZk+Yi3crNL27c72AQCAEwhLUTRihH184IC0YYORVozIyTHdAQAA0UdYiqIhQ+zjQEAqKzPTS6yEm1X6/HNn+wAAwCmEpSjKyLAWeh8tkRd5H+2UU0x3AABAbBCWomzwYPv4k0/M9BELqamh6zNnOtsHAABOIixFWcPNkr/+2kwfsVBdHbp+2WXO9gEAgJMIS1F22mn28cGD0ubNZnqJpoKC0PU77nC2DwAAnEZYirKGYSkQSIxTcZs2ha4/+KCzfSSaOXPmaMyYMcrJyZHH49GyZcuCXjNy5Eh5PB7bY/LkyQa6BYDkRFiKssxMqX17e6201Ewv0TJ6dOj62LHO9pGI9u/frxEjRujBY6TOa6+9Vlu3bq17PPPMMw51CABobbqBRDRggPTvf9ePFy8210s0vP9+6Prcuc72kYiuvPJKSdKGY2zIlZGRodzcXAc6AgA0xMxSDDS8jH71ajN9RMP06aHroW4cjNiZOXOmcnJyNGDAAM2YMUN79+4N+9rKykr5/X7bAwDQcoSlGGh42xOfT/ruOzO9ROoPfwhdd3MAdJvLL79cf//737VgwQLdc889euWVV3ThhReGff0DDzwgr9db9ygItzofANAkhKUYGDrUPg4EpBDrduPeCy+ErmdlOdtHopg5c6YyMzPrHh828U7L1157rUaPHq2BAwdq8uTJmj17tubNm6clS5aEfP1dd90ln89X9ygvL4/mbwMAkg5rlmKgWzepXTtp//762kcfSWPGmOupJa66KnSdszotM3HiRBUXF9eNu3Xr1qL3OeWUU5SamqrVq1frlBBbp6enpys9Pb3FfQIA7AhLMdCqlRWYVq2qr7nttidLl5ruIPFkZWUpKwrTcitWrFBVVZXy8vKi0BUA4FgISzEybJg9LK1fb66Xlgh3r7dAwNk+Et2uXbu0ceNGbdmyRZL0zTffSJJyc3OVm5urtWvXaubMmRo3bpxycnL01Vdf6bbbbtPgwYM1YsQIk60DQNJgzVKMnHyyfbxli/20XDzbudN0B8njtdde0+DBgzV+/HhJ0uTJkzV48GA9/fTTkqS0tDS9//77GjNmjPr166dbbrlF55xzjubNm6eUlBSTrQNA0vAEAswVxEJZmTRkiL22ZEnwjXbjkccTus5Xijv5/X55vV75fD5lZ2ebbgcAXIeZpRgpKrLWLh1twQIjrTTLkSOmOwAAIL4QlmIoJ8c+buKV4ka1DrOKjVklAECyIizFUL9+9jFXmAEA4D6EpRg67TT7uKLCTB9NxVolAACCEZZiqKjIPj50SFq3zkwvAACgZQhLMdTwajhJ+vRT5/toCmaVAAAIjbAUQ926SW3a2Gtu28kbAIBkR1iKseOOs4//8x8zfTSGWSUAAMIjLMVYwyviVq400wcAAGgZwlKMnXmmfez3x9dtT5hVSlwlJSUqLCxUUcMrDQAAzcLtTmJs8WJp+HB7bdEi60a78YCwlPi43QkARIaZpRhreENdyQpL8SBcUDpwwNk+AACIZ4SlGGvbVsrMtNfi/R5xbdua7gAAgPhBWHJAQYF9/OWXZvo4WrhZpbVrne0DAIB4R1hywA9/aB/H821PevUy3QEAAPGFsOSAhgu8KyulXbvM9CJJrcL8rf/rX872AQCAGxCWHDBiRHDt44+d76NWuCvdzj/f2T4AAHADwpIDTjwxeI3QvHlmemm4o3itX//a2T4AAHALwpIDUlOlrCx7zdT2AZs3h67/7nfO9gEAgFsQlhzSvbt9bOKqszFjQtfHj3e2DwAA3ISw5JBTT7WPd+92vod33w1df+MNZ/sAAMBNCEsOOf10+/jIEWnjRueOf/fdoesnnOBcDwAAuBFhySENtw+QpIULnTv+Aw+Erq9Z41wPAAC4EWHJIX37Bl8R59RtT2bPDl3ntiYAABwbYckhKSnB94j79FNnjn3xxaHr3DAXAIBjIyw5qOEeRxs2xP6Y69bF/hiITyUlJSosLFRRUZHpVgDA1TyBQLj9nBFtl14q/fOf9lqs//TD3TCXv/Xk4ff75fV65fP5lJ2dbbodAHAdZpYcdOaZwbXVq2N3vIMHY/feAAAkC8KSg0JtCvn++7E7XkZG6DqzSgAANB1hyUG9egXXTN0jDgAANA1hyUEeT/Dl+suWxe5YoTCrBABA8xCWHJabax+Hu7EtAACID4Qlhw0YYB8fOhT92R5mlQAAiB7CksPOOiu49s03zvcBAACahrDksFBh6Z13ovf+zCoBABBdhCWHDRoUXItmWAIAANFFWHJYq1ZS69b22pdfRue9mVUCACD6CEsGdOxoH3/3nZk+AADAsRGWDOjb1z6uqor8PcPNKnHLEwAAIkNYMuCMM4Jrsboirk2b2LwvAADJgrBkwNixwbU5c1r+fuFmldavb/l7AgAAC2HJgGHDgmvvvRf94xx/fPTfE+5RUlKiwsJCFRUVmW4FAFzNEwhwrZQJrVrZr1LLyZG2b2/++6SmStXVwfU5c6QLLmh5f0gcfr9fXq9XPp9P2dnZptsBANdhZsmQjAz7ePfulr1PqKAkEZQAAIgWwpIhxx1nHx850vz36NMndP3OO5v/XgAAIDTCkiHFxcG15l4Rt2ZN6PoDDzS/HwAAEBphyZBJk4JrL73U9M8Pd5rt7LNb1g8AAAiNBd6GHDoktW1rr/3wh1JZWdM+n1uboKlY4A0AkWFmyZBQm0V+/XXTPveRR0LX8/Nb3g8AAAiNmSWDWrcOXtjdlL8NZpXQHMwsAUBkmFkyqOENdZvio49C11vxNwkAQEzwI9agH/wguLZ8eeOfc/rpoest2XoAAAAcG2HJoEsvDa6VlIR//Y4dsesFAACExpolg/btk7Ky7LXu3aVvvw39etYqoSVYswQAkWFmyaDMzODa5s3O9wEAAMIjLMWZcGuPmFUCAMAMwpJhDTemBAAA8YWwZFjfvsG12bPtY2aV0BIlJSUqLCxUUVGR6VYAwNVY4G3Y//yPdMst9trJJ0tLl9aPCUuIBAu8ASAyhCXDqqqktDR7LSVFqq62fk1QQqQISwAQGU7DGZaaGlxjg0kAAOIHYSmOMasEAIB5hKU40Lp1cO39953vAwAABCMsxYFhw4Jro0eHfu3hw7HtBQAA2BGW4sCLLzb9taHWOAEAgNghLMWB7t2b9rqKitj2AQAAghGWXKRrV9MdAACQfAhLLrF4sekOAABIToSlOFFa2vjzQ4c60wcAALAjLMWJM84I/9zevc71AedUVVXpjjvu0KBBg9SuXTvl5+frqquu0pYtW2yv2717t6688kp5vV55vV5deeWV2rNnj6GuASD5EJbiyC9/GbqemelsH3DGgQMHtGTJEt1zzz1asmSJ5syZo1WrVmnixIm211122WVatmyZ3n77bb399ttatmyZrrzySkNdA0Dy4d5wcWb9eqlXL+vXDz0k3X672X7grLKyMg0ZMkTffvutunfvrpUrV6qwsFAff/yxiouLJUkff/yxhg0bpq+//lr9+vU75ntybzgAiEyIvaNhUs+e3M4kmfl8Pnk8HrVv316StHjxYnm93rqgJElDhw6V1+vVokWLQoalyspKVVZW1o39fn/sGweABMZpOCBOHDp0SHfeeacuu+yyuhmgiooKdenSJei1Xbp0UUWYjbceeOCBuvVNXq9XBQUFMe0bABIdYQlwyMyZM5WZmVn3+PDDD+ueq6qq0uTJk1VTU6M//vGPts/zhLijciAQCFmXpLvuuks+n6/uUV5eHt3fCAAkGU7DAQ6ZOHGi7XRat27dJFlB6ZJLLtH69es1f/5827qi3Nxcfffdd0HvtX37dnUNs0tpenq60tPTo9w9ACQvwhLgkKysLGVlZdlqtUFp9erV+uCDD9SpUyfb88OGDZPP59Onn36qIUOGSJI++eQT+Xw+DR8+vMnH9fl8QccGADQNV8MBhlRXV2vSpElasmSJ3njjDdtMUceOHZWWliZJGjt2rLZs2aJnnnlGknTdddepR48eev311430DQDJhrAEGLJhwwb17Nkz5HMffPCBRo4cKUnatWuXbrnlFr322muSrNN5Tz31VN0VcwCA2CIsAQAANIKr4QAAABpBWAIAAGgEYQkAAKARhCUAAIBGEJYAAAAaQVgCAABoBGEJAACgEYQlAACARhCWAAAAGkFYAgAAaARhCQAAoBH/H8k3e0wX9nr7AAAAAElFTkSuQmCC",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 10,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pruned=[point for point in sep_forward_hh_list if point[0]< 10.0]\n",
    "list_plot(pruned,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Perturb the initial point very slightly to display the instability in both directions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "T_back = back_point[0]\n",
    "hminus_back = back_point[1]\n",
    "hplus_back = back_point[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 12,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point_mod=[T_back,hminus_back,hplus_back - 1.31211060e-88 ]\n",
    "(sep_traj_mod,sep_mod_hh_list,number_pruned) =\\\n",
    "    calc_traj(point_mod,13.4)\n",
    "pruned=[point for point in sep_mod_hh_list if point[0]< 10.0]\n",
    "list_plot(pruned,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9uDR8eFzKapLaG2rn5uZaf3KbTZkyRaNGjdIPf/hDu0ux3BtvvKErr7xSN998szp37qyioiI999xzdpdlueHDh+u9995TWVmZJOmTTz7R+++/n9AbuiaznTt36sCBAxo5cmTdmNfr1VVXXaW1tfdfcSgu8G6kuXPnqnXr1rrvvvvsLiUpnDhxQjNmzND48eMdtTBjS0yZMkVTpkypW5SyuZp7P7QOHSL3lEoWhmHoV7/6lYYPH66+ffvaXY6lFi9erNLSUq1f39K7xDvTjh079Mwzz+hXv/qVfvOb32jdunW677775PV6ddttt9ldnmWmT5+uUCikSy+9VGlpaTp16pQeffRR3XLLLXaXZosDBw5Ikrp06VJvvEuXLtq9e7cdJcUNnaUoiouLlZWVVbeVlJToySef1AsvvCBPy+/86Qjnvgdr1qyp21dVVaVx48appqZGTz/9tI1VOkvt9UhNUVBwpnOUTEFJkn7xi1/o008/1X/913/ZXYqlysvLNXXqVP3xj39UmzZt7C7HFjU1NRowYIBmzZqloqIiTZ48WXfffbeeeeYZu0uz1Msvv6w//vGP+tOf/qTS0lK9+OKL+vd//3e9+OKLdpdmq3N/ThqG4fifnXSWohgzZowGDRpU9/Urr7yigwcPqnv37nVjp06d0v33368nnnhCu3btsqHKxDr3PejWrZukSFAaO3asdu7cqRUrVrimq9QSzfk3ItlvQnTvvffqjTfe0OrVq11xM+GzbdiwQQcPHtQVV1xRN3bq1CmtXr1a8+fPV2VlpdJqb2+forp27Sq/319vrHfv3vrzn/9sU0X2+PWvf60ZM2Zo3LhxkqR+/fpp9+7dmj17tm6//Xabq7Nefn6+pEiHqWvXrnXjBw8ebNBtchrCUhTZ2dnKPuvjSPfcc49Gjx5d7zHXXHONJk6cqDvvvNPq8ixx7nsgnQlK27Zt08qVK5WXl2dTdc5w113S8883/vE/+5mU7I06wzB077336rXXXtOqVavUs2dPu0uy3A9+8AN99tln9cbuvPNOXXrppZo+fXrKByVJGjZsWIMlI8rKytSjRw+bKrLH8ePH1eqcFVrT0tJSeumAWHr27Kn8/HwtX75cRafXITl58qRKSko0d+5cm6trGcJSI+Tl5TUIBunp6crPz9cll1xiU1XWqq6u1k033aTS0lK9+eabOnXqVN38dG5urjIyMmyuMLk0pZt06JDUsWPiaomnKVOm6E9/+pNef/11ZWdn130P+Hw+tW3b1ubqrJGdnd3gGq127dopLy/PNddu/fKXv9TQoUM1a9YsjR07VuvWrdPChQu1cOFCu0uz1OjRo/Xoo4+qe/fu6tOnjzZu3Kjf/e53mjRpkt2lJcyxY8f05Vk3nty5c6c2bdqk3Nxcde/eXdOmTdOsWbPUq1cv9erVS7NmzVJmZqbGjx9vY9VxYPOn8RzLbUsH7Ny505AUdVu5cqXd5Vkq1tIBTfmovxOZfQ8sWrTI7tJs5balAwzDMJYtW2b07dvX8Hq9xqWXXmosXLjQ7pIsFw6HjalTpxrdu3c32rRpY1x44YXGv/zLvxiVlZV2l5YwK1eujPpvwO23324YRmT5gJkzZxr5+fmG1+s1vv/97xufffaZvUXHgccwkv3qCCC51H4aLhQK1btmq7HdpJMnpfT0BBUHAIg7Pg0HtFBubuOC0uuvR3pKBCUAcBauWQJaoDEhqXXryG1GAADORFgCmqkx61IyyQ0Azsc0HNBEjbln6qBBBCUASBV0loAmaMy0GyEJAFILnSWgkQhKAOBOhCWgESJBKSjJLynQYP/KlQQlAEhVrLMEnEfDjlJYkk9SSFIOIQkAUhydJcDEsWPnn3ojKAFA6iMsAVGsXi2dcx/hBkIha2oBANiLsASc48knpauuMt/vknulAgBOIywBZ5kxQ5o2zXz/smXS//6vdfUAAOzHOkvAaTNnSnPnmu+vqorcuiQctq4mAID9CEuApOeek377W/P9XMgNAO7FNBxcb8sW6Z57zPcTlADA3QhLcLWqKqlPH/P9BCUAAGEJrpaRYb6PoAQAkAhLcLFYC04SlAAAtQhLcCWCEgCgsQhLcJ1YQamiwro6AADOQFiCq8QKSsXFUmam+f5gMCi/369AIBD/wgAASctjGEw6wB1iBaXBg6UPPmjcccLhsHw+n0KhkHJycuJTHAAgadFZgiuc76a4jQ1KAAD3ISwh5a1aJR07Zr6f3ioAIBbCElLe1Veb7yMoAQDOh7CElMYSAQCAliIsIWURlAAA8UBYQkoiKAEA4oWwhJQzcKD5vvHjrasDAJAaWGcJKSfRXSXWWQIAd6GzhJTC9BsAIN4IS0gZBCUAQCIQlpASCEoAgEQhLMHxnnzSfN/jj1tXBwAgNXGBNxzP6q4SF3gDgLvQWYKjWRmUgsGg/H6/AoFAfA8MAEhqdJbgWHZdp0RnCQDchc4SHOmGG8z3Ef8BAPFEWIIjLV0afbxTJ2vrAACkPsISHCfW9NvBg9bVAQBwB8ISHIX1lAAAViMswTF+9jPzfQQlAECiEJbgGAsWRB/v3t3aOgAA7kJYgiPEmn7bvdu6OgAA7kNYQtKL53VKd9xxhzweT71t8ODBLSsQAJDSWttdABDL9u3m+5p7ndKPfvQjLVq0qO7rjIyM5h0IAOAKhCUktYsvjj6eltb8Y3q9XuXn5zf/AAAAV2EaDkkr1vRbdXXzj7tq1Sp17txZ3/3ud3X33XfrIIszAQBi4N5wSEqdO0uHDkXf15Lv2JdffllZWVnq0aOHdu7cqYcffljV1dXasGGDvF5v1OdUVlaqsrKy7utwOKzCwkLuDQcALkFnCUnJLCgtX974YxQXFysrK6tuW7NmjX7yk59o1KhR6tu3r0aPHq23335bZWVl+stf/mJ6nNmzZ8vn89VthYWFTXw1AAAno7OEpBOvT78dPXpUX3/9dd3X3bp1U9u2bRs8rlevXvqnf/onTZ8+Pepx6CwBgLtxgTeSSjyXCcjOzlZ2dnbMx3zzzTcqLy9X165dTR/j9XpNp+gAAKmPaTgkjbIy833x6H8eO3ZMDzzwgD744APt2rVLq1at0ujRo9WxY0fdcMMNLT8BACAl0VlC0rjkkujj8fqUf1pamj777DP94Q9/0JEjR9S1a1ddffXVevnll8/bgQIAuBfXLCEpxHP6LdHC4bB8Ph/XLAGASzANB9uNGmW+L9mCEgDAfQhLsN1bb0Uff/FFa+sAACAapuFgKydNv9ViGg4A3IXOEmzjxKAEAHAfwhKSDkEJAJBMCEuwRayuEgAAyYSwBMsx/QYAcBLCEpIGQQkAkIwIS7CUWVepY0dr62iOYDAov9+vQCBgdykAAAuxdAAskyrTbywdAADuQmcJlti82Xyfk4ISAMB9CEuwRL9+0cfHjLG2DgAAmoqwhISLNf32+uvW1QEAQHMQlpBQCxaY72P6DQDgBIQlJNTPfhZ9/LHHrK0DAIDm4tNwSJhU+fTbufg0HAC4C50lJMRvfmO+z8lBCQDgPoQlJMTs2dHHV6ywtg4AAFqKsIS4izX9dvXV1tUBAEA8EJYQV3fdZb6P6TcAgBMRlhBXzz8ffXz3bmvrAAAgXghLiJtY02/du1tXBwAA8URYQlyMG2e+L1Wm34LBoPx+vwKBgN2lAAAsxDpLiAuzrtI330i5udbWkmisswQA7kJnCS0Wa/ot1YISAMB9CEtokTvuMN9HzxIAkAoIS2iRF1+MPn7kiLV1AACQKIQlNFus6Tefz7o6AABIJMISmuWnPzXfx/QbACCVEJbQLM8+G338m2+srQMAgEQjLKHJ+PQbAMBNCEtokkcfNd/H9BsAIBURltAkDz0UfXzbNmvrAADAKoQlNFqs6beLL7auDgAArERYQqO88or5PqbfAACpjLCERhk7Nvr4e+9ZWwcAAFYjLOG8Yk2//d3fWVeH3YLBoPx+vwKBgN2lAAAs5DEMJlFg7q9/lXr3jr7Prd854XBYPp9PoVBIOTk5dpcDAEgwOkuIySwoLVhgbR0AANiFzhJMxZp+c/N3DZ0lAHAXOkuIqrLSfJ+bgxIAwH0IS4iqTZvo45MnW1sHAAB2YxoODTD9FhvTcADgLnSW0GgEJQCAGxGWUI9ZV2nQIGvrAAAgWRCWUCfW9NuHH1pXBwAAyYSwhPNi+g0A4GaEJUgy7yrl5lpbBwAAyYawBGVlme/75hvr6gAAIBkRlqCKiujjTL8BAEBYcr1YF3UDAADCkqt9//vm++gqNRQMBuX3+xUIBOwuBQBgIVbwdjGzrhLfEbGxgjcAuAudJZdi+g0AgMYhLLnQvHnm++gqAQBQH9NwLmTWVQqFJGaVzo9pOABwFzpLLhNr+o2f+wAANERYcpGDB8330V8EACA6wpKLdOkSffwPf7C2DgAAnIRrllwi1vQb3wFNwzVLAOAudJZcjqAEAEBshCUXMOsqXXuttXUk2pIlS3TNNdeoY8eO8ng82rRpU4PHVFZW6t5771XHjh3Vrl07jRkzRnv27LGhWgCAUxCWUlys6be33rKuDitUVFRo2LBhmjNnjuljpk2bptdee02LFy/W+++/r2PHjunHP/6xTp06ZWGlAAAn4ZqlFOfGW5rs2rVLPXv21MaNG9W/f/+68VAopE6dOumll17ST37yE0nSvn37VFhYqLfeekvXXHNNo47PNUsA4C50llKYWVDKyrK2jmSxYcMGVVVVaeTIkXVjBQUF6tu3r9auXWv6vMrKSoXD4XobAMA9CEspqk8f831Hj1pXRzI5cOCAMjIy1KFDh3rjXbp00YEDB0yfN3v2bPl8vrqtsLAw0aUCAJIIYSlFbdkSfTxVpt+Ki4uVlZVVt61Zs6bZxzIMQ54YF3c9+OCDCoVCdVt5eXmzzwUAcJ7WdheA+It1UXeqGDNmjAYNGlT3dbdu3c77nPz8fJ08eVKHDx+u1106ePCghg4davo8r9crr9fbsoIBAI5FZynFPPaY+b5U6SpJUnZ2ti6++OK6rW3btud9zhVXXKH09HQtX768bmz//v3avHlzzLAEAHA3OkspZvr06OPHj1tbhx2+/fZbffXVV9q3b58k6YsvvpAU6Sjl5+fL5/Pprrvu0v3336+8vDzl5ubqgQceUL9+/fTDH/7QztIBAEmMzlIKiTX91ojGi+O98cYbKioq0qhRoyRJ48aNU1FRkRYsWFD3mMcff1zXX3+9xo4dq2HDhikzM1PLli1TWlqaXWUDAJIc6yyliKNHJbMlf/gTji/WWQIAd6GzlCLMfmY//7y1dQAAkGroLKWAWNNv/OnGH50lAHAXOkspjKAEAEDLEZYczqyrVFRkbR0AAKQqwpKDxZoBKi21rg4AAFIZYcnBzO7xxvQbAADxQ1hyKDfc0gQAgGRAWHKgyZPN99FVSpxgMCi/369AIGB3KQAAC7F0gAOZdZX4k7QGSwcAgLvQWXIYpt8AALAWYclB9u8330dXCQCAxCAsOUhBQfTxFSusrQMAADchLDlErOm3q6+2rg4AANyGsORwTL8BAJBYhCUHMOsqff/71tYBAIAbEZaSXKxPppeUWFcHAABuRVhKctzSBAAAexGWkhhrKgEAYD/CUpL6538230dXCQAA63C7kyTFLU2SF7c7AQB3obOUhJh+AwAgeRCWkozZBd0SXSW7BYNB+f1+BQIBu0sBAFiIabgkY9ZVevVV6cYbra0F0TENBwDuQmcpicSafiMoAQBgD8KSA9D7AwDAPoSlJGHWVbr8cmvrAAAA9RGWkkDPnub7Nm2yrg4AANAQYSkJ7NoVfZzpNwAA7EdYshlrKgEAkNwISzZ64QXzfXSVAABIDoQlG915Z/Tx48etrQMAAJgjLNkk1vRb27bW1QEAAGIjLCUZpt8AAEguhCUbmHWVHnrI2joAAMD5cW84i8WafuNPwhm4NxwAuAudpSRBUEp+wWBQfr9fgUDA7lIAABais2Qhs66SzycdOWJtLWg+OksA4C50lixyyy3m+whKAAAkL8KSRRYvjj5OXw8AgORGWLIAtzQBAMC5CEsJFmuKja4SAADJj7CUYB06RB9fu9baOgAAQPMQlhIo1vTbkCHW1QEAAJqPsGQDpt8AAHAOwlKCmHUmw/keAAALYklEQVSVBgywtg4AANAyhKUEuPRS830bNlhXBwAAaDnCUgJ88UX0cabfAABwHsJSnLGmEgAAqYWwFEerVpnvo6sEAIAzEZbi6Oqro4+Hw9bWAQAA4oewFCexpt+ys62rA4kTDAbl9/sVCATsLgUAYCGPYTBBFA9mYYl3N/WEw2H5fD6FQiHl5OTYXQ4AIMHoLMWBWVC64w5LywAAAAlAWGqhWI2FRYusqwMAACQGYamFjh6NPs70GwAAqYGw1AJm02+stQQAQOogLDXTvHnm+2pqrKsDAAAkFp+GayY+/eZefBoOANyFzlIzMM0GAIB7EJbiiK4SAACph7DURGZdpVmzrK0DDS1ZskTXXHONOnbsKI/Ho02bNjV4zIgRI+TxeOpt48aNs6FaAIBTEJaaINb024MPWlcHoquoqNCwYcM0Z86cmI+7++67tX///rrt2WeftahCAIATtba7gFTA9FtymDhxoiRp165dMR+XmZmp/Px8CyoCAKQCOkuNZNZV4ia5zlNcXKyOHTuqT58+euCBB3TUbGXR0yorKxUOh+ttAAD3oLPUCL/8pfk+fm46y6233qqePXsqPz9fmzdv1oMPPqhPPvlEy5cvN33O7Nmz9a//+q8WVgkASCass9QIrKmUfIqLizV58uS6r99++21973vfkxSZhuvZs6c2btyo/v37xzzOhg0bdOWVV2rDhg0aMGBA1MdUVlaqsrKy7utwOKzCwkLWWQIAl6CzdB6sqZScxowZo0GDBtV93a1bt2YdZ8CAAUpPT9e2bdtMw5LX65XX623W8QEAzkdYiqG62nwfXSV7ZWdnKzsOF4x9/vnnqqqqUteuXeNQFQAgFRGWYkhPjz6+eLG1daBxvv32W3311Vfat2+fJOmLL76QJOXn5ys/P1/bt29XcXGxrrvuOnXs2FFbtmzR/fffr6KiIg0bNszO0gEASYxrlkzEmn7jHUtOL7zwgu68884G4zNnztQjjzyi8vJyTZgwQZs3b9axY8dUWFioUaNGaebMmcrNzW30ebg3HAC4C2HJBBd1wwxhCQDchXWWojALSlzWAgCA+xCWznHLLeb7Tl8KAwAAXISwdA6zi7eZfgMAwJ0IS2dhTSUAAHAuwtJpFRXm++gqAQDgXoSl07Kyoo+vWGFtHQAAILkQlhR7+u3qq62rAwAAJB/CUgxMv+FswWBQfr9fgUDA7lIAABZy/aKUZl2liy6SvvzS2lrgDCxKCQDu4urO0ujR5vsISgAAQHJ5WHrzzejj7u61AQCAs7k2LLGmEgAAaAxXhqVQyHwfXSUAAHA2V4al9u2jj3/8sbV1AACA5Oe6sBRr+u2KK6yrAwAAOIPrwpIZpt8AAEA0rgpLZl2l3r2trQMAADiHa8LStdea79uyxbo6AACAs7gmLL3zTvRxpt8AAEAsrghLrKkEAACaK+XDEmsqAQCAlkj5sGS2ptKnn1pbB5wvGAzK7/crEAjYXQoAwEIew0jd/kqs6bfUfdVItHA4LJ/Pp1AopJycHLvLAQAkWMp3lqIhKAEAgMZK2bBk1lXy+62tAwAAOFtKhqUf/9h83+efW1cHAABwvpQMS3/5S/Rxpt8AAEBTpVxYYk0lAAAQTykVllhTCQAAxFtKhSWzNZU2bbK2DgAAkDpSJizFmn67/HLr6gAAAKklZcKSGabfAABAS6REWDLrKvXubW0dAAAg9Tg+LI0ebb5vyxbr6gAAAKnJ8WHpzTejjzP9BgAA4sHRYYk1lWClYDAov9+vQCBgdykAAAt5DMOZPZhjx6Ts7Oj7nPmK4BThcFg+n0+hUEg5OTl2lwMASDDHdpbMgtLatdbWAQAAUpsjw1Ks6bchQ6yrAwAApD5HhiUzTL8BAIB4c1xYMusqXXihtXUAAAB3cFRYuu02833bt1tXBwAAcA9HfRrOrKvknFeAVMCn4QDAXRzTWWJNJQAAYAdHhKWTJ8330VUCAACJ5Iiw5PVGH3/7bWvrAAAA7pP0YSnW9NuPfmRdHQAAwJ2SPiyZYfoNAABYIanDkllXKT/f2joAAIB7JW1Yuu8+833791tXBwAAcLekDUu//330cabfYJdgMCi/369AIGB3KQAACyXlopSxLupOvmrhNixKCQDukrSdpWgISgAAwGpJF5bMukovvWRtHQAAAFKSTcMx/QYnYBoOANwl6TpL0RCUAACAXZImLJl1lXw+a+sAAAA4W1KEpTlzzPcdOWJdHQAAAOdKimuWzLpK9lcGNMQ1SwDgLrZ3lmJd1A0AAGA328OSGbpKaIqqqipNnz5d/fr1U7t27VRQUKDbbrtN+/btq/e4w4cPa+LEifL5fPL5fJo4caKOMNcLAIjB1rBk1lWaO9faOuB8x48fV2lpqR5++GGVlpZqyZIlKisr05gxY+o9bvz48dq0aZPeeecdvfPOO9q0aZMmTpxoU9UAACew7ZqlzEzpb3+Lvo+uEuJh/fr1GjhwoHbv3q3u3btr69at8vv9+vDDDzVo0CBJ0ocffqghQ4bor3/9qy655JJGHZdrlgDAXWzrLBGUkGihUEgej0ft27eXJH3wwQfy+Xx1QUmSBg8eLJ/Pp7Vr15oep7KyUuFwuN4GAHAPW8KS2fRberq1dSB1nThxQjNmzND48ePruj8HDhxQ586dGzy2c+fOOnDggOmxZs+eXXeNk8/nU2FhYcLqBgAkH8vD0qJF5vtOnrSuDjhbcXGxsrKy6rY1a9bU7auqqtK4ceNUU1Ojp59+ut7zPFGSumEYUcdrPfjggwqFQnVbeXl5/F4IACDptbb6hJMmRR9n+g1NMWbMmHrTad26dZMUCUpjx47Vzp07tWLFinrXFOXn5+vrr79ucKxDhw6pS5cupufyer3yer1xrB4A4CSWhiXWVEK8ZGdnKzs7u95YbVDatm2bVq5cqby8vHr7hwwZolAopHXr1mngwIGSpI8++kihUEhDhw61rHYAgLNY+mk4VupGolRXV+vGG29UaWmp3nzzzXqdotzcXGVkZEiSrr32Wu3bt0/PPvusJOmee+5Rjx49tGzZskafi0/DAYC7WHbNkllQuu8+qypAKtuzZ4/eeOMN7dmzR/3791fXrl3rtrM/6VZcXKx+/fpp5MiRGjlypC677DK99NJLNlYOAEh2lnSWCgqk/fuj76OrBKehswQA7mJJZ4mgBAAAnCrhYYmLugEAgJMlNCyVlZnvo6sEAACcIKFhyexWWxUViTwrAABA/CQsLMW6J2lmZqLOCgAAEF8JC0tmU3BMvwEAACdJSFiqqoo+fsEFiTgbAABA4iQkLJ1eLLmBnTsTcTbAGsFgUH6/X4FAwO5SAAAWSsiilNzWBKmMRSkBwF0su90JQQkAADiRZWEJAADAiQhLAAAAMRCWAAAAYiAsAQAAxEBYAgAAiCEhYalVlKOeOpWIMwEAACRWQsJS+/YNx0pKEnEmAACAxEpIWOrTp+HYa68l4kwAAACJlZCwdMMNDcfWrUvEmQAAABIrIWHp1lsbjpWXJ+JMAAAAiZWQsNS5c8OxI0cScSYAAIDEsmzpgMpKq84EJEYwGJTf71cgELC7FACAhTyGkZhb3Ho8Dce4mS5SQTgcls/nUygUUk5Ojt3lAAASLGGdpbS0RB0ZAADAOgkLS5dcUv9rrzdRZwIAAEichIWll1+u31165JFEnQkAACBxWifqwH37Stu3SytXSr16ScOGJepMAAAAiZOwC7yBVMUF3gDgLpYtHQAAAOBEhCUAAIAYCEsAAAAxEJYAAABi4AJvoIkMw9DRo0eVnZ0tT7Sl6gEAKYWwBAAAEAPTcAAAADEQlgAAAGIgLAEAAMRAWAIAAIiBsAQAABADYQkAACAGwhIAAEAMhCUAAIAYCEsAAAAxEJYAAABi+H/s14n51EKcugAAAABJRU5ErkJggg==",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 13,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point_mod_2=[T_back,hminus_back,hplus_back - 1.31211061e-88]\n",
    "(sep_traj_mod_2,sep_mod_2_hh_list,number_pruned_2) =\\\n",
    "    calc_traj(point_mod_2,13.4)\n",
    "pruned=[point for point in sep_mod_2_hh_list if point[0]< 10.0]\n",
    "list_plot(pruned,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "The difference in the initial $h_{+}$ values is $O(10^{-96})$.  This is much smaller than the tolerance parameters of the ode solver."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Define a function to calculate the double expansion $e^S(x,y)$ to $O(x^N y^N)$\n",
    "and a function to use $e^S(x,y)$ to calculate $c$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "Rxy.<xx,yy> = R[]\n",
    "def calc_expS(N):\n",
    "    j_size = N+1\n",
    "    k_size = N+1\n",
    "    zklist = [0.0 for k in range(k_size)]\n",
    "    Sjk = [[0.0 for k in range(k_size)] for j in range(j_size)]\n",
    "    Sjk[0][0] = 1.0\n",
    "    if j_size > 1:\n",
    "        Sjk[1][0] = 1.0\n",
    "    for k in range(1,k_size):\n",
    "        for j in range(1,j_size):\n",
    "            Sjk[j][k] = ((2-j-(1+nu)*k)*Sjk[j-1][k] \\\n",
    "                         + (lambdaminus*j+(1-nu)*(k-1))*Sjk[j][k-1]) \\\n",
    "                        /(j+(2+nu)*k)\n",
    "    expS = 0.0\n",
    "    for j in range(j_size):\n",
    "        for k in range(k_size):\n",
    "            expS+= Sjk[j][k]*xx^j*yy^k\n",
    "    return(expS)\n",
    "def calc_c(point,expS):\n",
    "    hminus_val = point[0]\n",
    "    hplus_val = point[1]\n",
    "    x_val = 1.0/hminus_val^2\n",
    "    y_val = (hplus_val*hminus_val+1.0)*x_val\n",
    "    expS_val = expS(xx=x_val,yy=y_val)\n",
    "    c_val = expS_val*y_val*(x_val^(-2.0-nu))\n",
    "    return(c_val)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "For one of the two trajectories bracketing the separatrix, calculate $c$ at the earliest point for several values of $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "cvals={}\n",
    "for N in range(10):\n",
    "    expS = calc_expS(N)\n",
    "    cvals[N] = calc_c(point_mod[1:],expS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 0.28445788665466583381471805147876193750348369531743315490387388017381802005436613900703825930517289597954613816749432817,\n",
       " 1: 0.28445788863675554418877978914259637544544616653289475791447457114119762673701745040770625196759798492977030445020689204,\n",
       " 2: 0.28445788863675554418877978914259637544053557077520960232297491369849384079444308496582665196457907472588138752667272458,\n",
       " 3: 0.28445788863675554418877978914259637544053557079029671698615895337795517756130003494036406043263051121782074883638954280,\n",
       " 4: 0.28445788863675554418877978914259637544053557079029671693057357826578273290504777101635027895311635213621996382967154836,\n",
       " 5: 0.28445788863675554418877978914259637544053557079029671693057357849535473125067261485784686777059223949814030804694403173,\n",
       " 6: 0.28445788863675554418877978914259637544053557079029671693057357849535473022464798120993511983711278571617736358154813162,\n",
       " 7: 0.28445788863675554418877978914259637544053557079029671693057357849535473022464798606926241540811483036562653646955135020,\n",
       " 8: 0.28445788863675554418877978914259637544053557079029671693057357849535473022464798606926239134773811471851460068024544137,\n",
       " 9: 0.28445788863675554418877978914259637544053557079029671693057357849535473022464798606926239134773823811940895094995129935}"
      ]
     },
     "execution_count": 16,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cvals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Check that the expansion improves with $N$ for small $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 17,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NN=0\n",
    "logdevs = [[N,log(abs(cvals[N]-cvals[9]),10)] for N in cvals if N >=NN and N < 9]\n",
    "list_plot(logdevs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Evaluate the deviation in $c$ along the early part of the separatrix, i.e., small $T$, large $h_{-}$ to check the accuracy of the expansion of $e^s(x,y)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 18,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hminus_min = 30\n",
    "expS = calc_expS(9)\n",
    "Tc_list=[[point[0],calc_c(point[1:],expS)-cvals[9]] \n",
    "         for point in sep_traj_mod if point[1]>hminus_min]\n",
    "list_plot(Tc_list,axes_labels=[r\"$T$\",r\"$\\Delta c$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### check numerical precision"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
      ]
     },
     "execution_count": 19,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nu*(1+nu)-0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5490367659397273091272122533145407390280876768195163461951051765377665392390743859450728937108058387632230170030217870e-120"
      ]
     },
     "execution_count": 20,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lambdaplus*lambdaminus-1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### check large T expansion of separatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.9109714707410239459590521448978416357858977397635351659452142136058193323590324881683916140189492021067141957989139353e10*y^39 - 1.2371689146222568392662362600849397377664170507382153409043180489598123902677680903280798725102520982241349403752203940e10*y^37 + 2.0918528906674824373038519446503936596269825857957406873269514622352053620857438680864347427268892151141347116033286798e9*y^35 - 3.4192511731384801012290318940091859706049161508458237596236575505396044057852560930905279090897117784985942695370246068e8*y^33 + 5.7801148446586037557755313033172040022643511902463346288739693965031963479205623926760244535217772375883483574423198136e7*y^31 - 1.0582397791770538048771035564399633674067094158777555783081640148113461085373497061819114286825017841070985975671496604e7*y^29 + 2.2082062131082423016774882521181040588803694045790884526378724057795686447147031380724858115814498367519576002969360312e6*y^27 - 570053.89406978727818174708041765976050801091928654064619561929597054758230339573988144527471369319219570431166180577056*y^25 + 218938.04630259429567173498079193690771655290796280525505548848219249713180618876230454194973335963065188088530758932253*y^23 - 2.7048765579483540656291006431062453471417056851230520698307813153751529101669157691566655152089325758793545908391846767e6*y^21 + 2.0303534698525193786192196446644348374588950797608764652848482569915813583114395791459259432446274289837316728462007167e-115*y^19 + 2.5379418373156492232740245558305435468236188497010955816060603212394766978892994739324074290557842862296645910577508958e-116*y^17 - 6.3448545932891230581850613895763588670590471242527389540151508030986917447232486848310185726394607155741614776443772396e-117*y^15 + 3.9655341208057019113656633684852242919119044526579618462594692519366823404520304280193866078996629472338509235277357748e-118*y^13"
      ]
     },
     "execution_count": 21,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_tilde_coeffs = 10\n",
    "calc_fv_polys()\n",
    "test_ode = y*(0.5*y^2*f_poly.derivative(y)-f_poly*v_poly-f_poly^2-1)+f_poly\n",
    "test_ode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### check accuracy of large T expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 1001  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 22,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_tilde_coeffs =100\n",
    "calc_fv_polys()\n",
    "[sep_traj_list_1,sep_hh_list_1] = separatrix(100,101)\n",
    "#list_plot(sep_hh_list_1)\n",
    "diff_list = [[point[0],point[1]-s_point(point[0])[1]] for point in sep_traj_list_1]\n",
    "list_plot(diff_list)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### Try the large $T$ expansion for larger $T_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 4251  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 23,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj,sep_hh_list)=separatrix(30,34.25)\n",
    "list_plot(sep_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0  points pruned 2651  points\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 24,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sep_traj,sep_hh_list)=separatrix(50,52.65)\n",
    "list_plot(sep_hh_list,axes_labels=[r\"$h_{-}$\",r\"$h_{+}$\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath (stable)",
   "language": "sagemath",
   "metadata": {
    "cocalc": {
     "description": "Open-source mathematical software system",
     "priority": 10,
     "url": "https://www.sagemath.org/"
    }
   },
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}