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    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats=['svg']\n",
    "import matplotlib.pyplot as plt\n",
    "from GeneralModel import GeneralModel, DataStore"
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    "# Demonstration of a fast ODE simulator\n",
    "\n",
    "+ The package $GeneralModel$ provides utility for quick prototyping and plotting of ODE's. \n",
    "+ There are some pre-defined models such as $SIR$ and $VanderPol$, however you can define your own model as well. \n",
    "+ The package provides time series and phase plot functionality. \n",
    "+ The parameters of the model are passed to the object as keyword arguments."
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    "# SIR Model\n",
    "\n",
    "The SIR model is a compartmental model that represents spread of an infectious disease through a population of size $N$.\n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "    \\dot{S} &= - \\beta S I \\\\\n",
    "    \\dot{I} &= \\beta I S - \\gamma R \\\\\n",
    "    \\dot{R} &= \\gamma R\n",
    "\\end{align*}\n",
    "\n",
    "There are only two parameters, $\\beta$ and $\\gamma$ that govern the system. $S$, $I$, and $R$ stand for Susceptible, Infected, and Recovered populations respectively."
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   "source": [
    "SIR = GeneralModel(Name=\"SIR\", tstart=0, tend=10, dt=0.1, beta=4, gamma=1, N=1)\n",
    "SIR.Initialize([0.9,0.1,0])\n",
    "SIR.Simulate(Flow=\"SIR\")\n",
    "plt.rcParams[\"figure.figsize\"] = (13, 4)\n",
    "SIR.PlotState(states={0:\"S\",1:\"I\",2:\"R\"},colors=[\"b\",\"r\",\"g\"],legend=[\"Susceptible\",\"Infected\",\"Recovered\"])"
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  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "DS = DataStore(SIR)\n",
    "DS.SaveState(\"mytest\")"
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  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
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       "      <td>0.1</td>\n",
       "      <td>0.859929</td>\n",
       "      <td>0.128686</td>\n",
       "      <td>0.011386</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.2</td>\n",
       "      <td>0.811393</td>\n",
       "      <td>0.162698</td>\n",
       "      <td>0.025909</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.3</td>\n",
       "      <td>0.754512</td>\n",
       "      <td>0.201410</td>\n",
       "      <td>0.044079</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.4</td>\n",
       "      <td>0.690344</td>\n",
       "      <td>0.243357</td>\n",
       "      <td>0.066299</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.5</td>\n",
       "      <td>0.620949</td>\n",
       "      <td>0.286267</td>\n",
       "      <td>0.092785</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.6</td>\n",
       "      <td>0.549175</td>\n",
       "      <td>0.327331</td>\n",
       "      <td>0.123495</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.7</td>\n",
       "      <td>0.478193</td>\n",
       "      <td>0.363709</td>\n",
       "      <td>0.158098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.8</td>\n",
       "      <td>0.410923</td>\n",
       "      <td>0.393074</td>\n",
       "      <td>0.196003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.9</td>\n",
       "      <td>0.349573</td>\n",
       "      <td>0.413999</td>\n",
       "      <td>0.236428</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.295427</td>\n",
       "      <td>0.426073</td>\n",
       "      <td>0.278500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>1.1</td>\n",
       "      <td>0.248888</td>\n",
       "      <td>0.429760</td>\n",
       "      <td>0.321353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.2</td>\n",
       "      <td>0.209685</td>\n",
       "      <td>0.426118</td>\n",
       "      <td>0.364197</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1.3</td>\n",
       "      <td>0.177134</td>\n",
       "      <td>0.416500</td>\n",
       "      <td>0.406366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1.4</td>\n",
       "      <td>0.150356</td>\n",
       "      <td>0.402310</td>\n",
       "      <td>0.447334</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1.5</td>\n",
       "      <td>0.128441</td>\n",
       "      <td>0.384848</td>\n",
       "      <td>0.486710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>1.6</td>\n",
       "      <td>0.110541</td>\n",
       "      <td>0.365234</td>\n",
       "      <td>0.524225</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>1.7</td>\n",
       "      <td>0.095911</td>\n",
       "      <td>0.344378</td>\n",
       "      <td>0.559711</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>1.8</td>\n",
       "      <td>0.083925</td>\n",
       "      <td>0.322995</td>\n",
       "      <td>0.593080</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>1.9</td>\n",
       "      <td>0.074069</td>\n",
       "      <td>0.301624</td>\n",
       "      <td>0.624307</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>2.0</td>\n",
       "      <td>0.065927</td>\n",
       "      <td>0.280657</td>\n",
       "      <td>0.653416</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>2.1</td>\n",
       "      <td>0.059167</td>\n",
       "      <td>0.260373</td>\n",
       "      <td>0.680461</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>2.2</td>\n",
       "      <td>0.053523</td>\n",
       "      <td>0.240958</td>\n",
       "      <td>0.705519</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>2.3</td>\n",
       "      <td>0.048786</td>\n",
       "      <td>0.222529</td>\n",
       "      <td>0.728685</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>2.4</td>\n",
       "      <td>0.044788</td>\n",
       "      <td>0.205152</td>\n",
       "      <td>0.750060</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>2.5</td>\n",
       "      <td>0.041395</td>\n",
       "      <td>0.188853</td>\n",
       "      <td>0.769752</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>2.6</td>\n",
       "      <td>0.038502</td>\n",
       "      <td>0.173631</td>\n",
       "      <td>0.787867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>2.7</td>\n",
       "      <td>0.036021</td>\n",
       "      <td>0.159465</td>\n",
       "      <td>0.804514</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>2.8</td>\n",
       "      <td>0.033885</td>\n",
       "      <td>0.146320</td>\n",
       "      <td>0.819795</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>2.9</td>\n",
       "      <td>0.032038</td>\n",
       "      <td>0.134152</td>\n",
       "      <td>0.833811</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>70</th>\n",
       "      <td>7.0</td>\n",
       "      <td>0.017928</td>\n",
       "      <td>0.003132</td>\n",
       "      <td>0.978940</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>71</th>\n",
       "      <td>7.1</td>\n",
       "      <td>0.017907</td>\n",
       "      <td>0.002854</td>\n",
       "      <td>0.979239</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>7.2</td>\n",
       "      <td>0.017888</td>\n",
       "      <td>0.002601</td>\n",
       "      <td>0.979511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>7.3</td>\n",
       "      <td>0.017870</td>\n",
       "      <td>0.002370</td>\n",
       "      <td>0.979760</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>7.4</td>\n",
       "      <td>0.017854</td>\n",
       "      <td>0.002160</td>\n",
       "      <td>0.979986</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>7.5</td>\n",
       "      <td>0.017839</td>\n",
       "      <td>0.001969</td>\n",
       "      <td>0.980193</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>7.6</td>\n",
       "      <td>0.017825</td>\n",
       "      <td>0.001794</td>\n",
       "      <td>0.980381</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>7.7</td>\n",
       "      <td>0.017813</td>\n",
       "      <td>0.001635</td>\n",
       "      <td>0.980552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>7.8</td>\n",
       "      <td>0.017802</td>\n",
       "      <td>0.001490</td>\n",
       "      <td>0.980708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>7.9</td>\n",
       "      <td>0.017792</td>\n",
       "      <td>0.001358</td>\n",
       "      <td>0.980850</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>8.0</td>\n",
       "      <td>0.017783</td>\n",
       "      <td>0.001237</td>\n",
       "      <td>0.980980</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>8.1</td>\n",
       "      <td>0.017774</td>\n",
       "      <td>0.001128</td>\n",
       "      <td>0.981098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>8.2</td>\n",
       "      <td>0.017767</td>\n",
       "      <td>0.001028</td>\n",
       "      <td>0.981206</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>8.3</td>\n",
       "      <td>0.017760</td>\n",
       "      <td>0.000936</td>\n",
       "      <td>0.981304</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>8.4</td>\n",
       "      <td>0.017753</td>\n",
       "      <td>0.000853</td>\n",
       "      <td>0.981393</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>8.5</td>\n",
       "      <td>0.017748</td>\n",
       "      <td>0.000778</td>\n",
       "      <td>0.981475</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>8.6</td>\n",
       "      <td>0.017742</td>\n",
       "      <td>0.000709</td>\n",
       "      <td>0.981549</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>8.7</td>\n",
       "      <td>0.017738</td>\n",
       "      <td>0.000646</td>\n",
       "      <td>0.981617</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>8.8</td>\n",
       "      <td>0.017733</td>\n",
       "      <td>0.000588</td>\n",
       "      <td>0.981678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>8.9</td>\n",
       "      <td>0.017729</td>\n",
       "      <td>0.000536</td>\n",
       "      <td>0.981735</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>9.0</td>\n",
       "      <td>0.017726</td>\n",
       "      <td>0.000489</td>\n",
       "      <td>0.981786</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>9.1</td>\n",
       "      <td>0.017722</td>\n",
       "      <td>0.000445</td>\n",
       "      <td>0.981832</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>9.2</td>\n",
       "      <td>0.017719</td>\n",
       "      <td>0.000406</td>\n",
       "      <td>0.981875</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>9.3</td>\n",
       "      <td>0.017716</td>\n",
       "      <td>0.000370</td>\n",
       "      <td>0.981914</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>9.4</td>\n",
       "      <td>0.017714</td>\n",
       "      <td>0.000337</td>\n",
       "      <td>0.981949</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>9.5</td>\n",
       "      <td>0.017712</td>\n",
       "      <td>0.000307</td>\n",
       "      <td>0.981981</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>9.6</td>\n",
       "      <td>0.017710</td>\n",
       "      <td>0.000280</td>\n",
       "      <td>0.982011</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>9.7</td>\n",
       "      <td>0.017708</td>\n",
       "      <td>0.000255</td>\n",
       "      <td>0.982037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>9.8</td>\n",
       "      <td>0.017706</td>\n",
       "      <td>0.000232</td>\n",
       "      <td>0.982062</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>9.9</td>\n",
       "      <td>0.017704</td>\n",
       "      <td>0.000212</td>\n",
       "      <td>0.982084</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    Time       x_1       x_2       x_3\n",
       "0    0.0  0.900000  0.100000  0.000000\n",
       "1    0.1  0.859929  0.128686  0.011386\n",
       "2    0.2  0.811393  0.162698  0.025909\n",
       "3    0.3  0.754512  0.201410  0.044079\n",
       "4    0.4  0.690344  0.243357  0.066299\n",
       "5    0.5  0.620949  0.286267  0.092785\n",
       "6    0.6  0.549175  0.327331  0.123495\n",
       "7    0.7  0.478193  0.363709  0.158098\n",
       "8    0.8  0.410923  0.393074  0.196003\n",
       "9    0.9  0.349573  0.413999  0.236428\n",
       "10   1.0  0.295427  0.426073  0.278500\n",
       "11   1.1  0.248888  0.429760  0.321353\n",
       "12   1.2  0.209685  0.426118  0.364197\n",
       "13   1.3  0.177134  0.416500  0.406366\n",
       "14   1.4  0.150356  0.402310  0.447334\n",
       "15   1.5  0.128441  0.384848  0.486710\n",
       "16   1.6  0.110541  0.365234  0.524225\n",
       "17   1.7  0.095911  0.344378  0.559711\n",
       "18   1.8  0.083925  0.322995  0.593080\n",
       "19   1.9  0.074069  0.301624  0.624307\n",
       "20   2.0  0.065927  0.280657  0.653416\n",
       "21   2.1  0.059167  0.260373  0.680461\n",
       "22   2.2  0.053523  0.240958  0.705519\n",
       "23   2.3  0.048786  0.222529  0.728685\n",
       "24   2.4  0.044788  0.205152  0.750060\n",
       "25   2.5  0.041395  0.188853  0.769752\n",
       "26   2.6  0.038502  0.173631  0.787867\n",
       "27   2.7  0.036021  0.159465  0.804514\n",
       "28   2.8  0.033885  0.146320  0.819795\n",
       "29   2.9  0.032038  0.134152  0.833811\n",
       "..   ...       ...       ...       ...\n",
       "70   7.0  0.017928  0.003132  0.978940\n",
       "71   7.1  0.017907  0.002854  0.979239\n",
       "72   7.2  0.017888  0.002601  0.979511\n",
       "73   7.3  0.017870  0.002370  0.979760\n",
       "74   7.4  0.017854  0.002160  0.979986\n",
       "75   7.5  0.017839  0.001969  0.980193\n",
       "76   7.6  0.017825  0.001794  0.980381\n",
       "77   7.7  0.017813  0.001635  0.980552\n",
       "78   7.8  0.017802  0.001490  0.980708\n",
       "79   7.9  0.017792  0.001358  0.980850\n",
       "80   8.0  0.017783  0.001237  0.980980\n",
       "81   8.1  0.017774  0.001128  0.981098\n",
       "82   8.2  0.017767  0.001028  0.981206\n",
       "83   8.3  0.017760  0.000936  0.981304\n",
       "84   8.4  0.017753  0.000853  0.981393\n",
       "85   8.5  0.017748  0.000778  0.981475\n",
       "86   8.6  0.017742  0.000709  0.981549\n",
       "87   8.7  0.017738  0.000646  0.981617\n",
       "88   8.8  0.017733  0.000588  0.981678\n",
       "89   8.9  0.017729  0.000536  0.981735\n",
       "90   9.0  0.017726  0.000489  0.981786\n",
       "91   9.1  0.017722  0.000445  0.981832\n",
       "92   9.2  0.017719  0.000406  0.981875\n",
       "93   9.3  0.017716  0.000370  0.981914\n",
       "94   9.4  0.017714  0.000337  0.981949\n",
       "95   9.5  0.017712  0.000307  0.981981\n",
       "96   9.6  0.017710  0.000280  0.982011\n",
       "97   9.7  0.017708  0.000255  0.982037\n",
       "98   9.8  0.017706  0.000232  0.982062\n",
       "99   9.9  0.017704  0.000212  0.982084\n",
       "\n",
       "[100 rows x 4 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DS.LoadData('mytest.csv')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Van der Pol Oscillator\n",
    "\n",
    "The Van der Pol oscillator is the classic example of a nonlinear system with a limit cycle. It is usually shown in the following form. \n",
    "\n",
    "\\begin{align*}\n",
    "    \\ddot{x} = \\mu (x^2 - 1)\\dot{x} + x = 0\n",
    "\\end{align*}\n",
    "\n",
    "We can split this into two first order equations as follows:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\dot{x} &= y \\\\\n",
    "    \\dot{y} &= -x - \\mu (x^2 - 1)y\n",
    "\\end{align*}\n",
    "\n",
    "The parameter $\\mu$ determines the existance and general behavior of the limit cycle."
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   "source": [
    "VDP = GeneralModel(Name=\"VanderPol\", tstart=0, tend=50, dt=0.01, mu=0.5) \n",
    "VDP.Initialize([0.9,0.1]) \n",
    "VDP.Simulate(Flow=\"VanderPol\") \n",
    "plt.rcParams[\"figure.figsize\"] = (13, 4)\n",
    "VDP.PlotState(states={0:\"x\",1:\"y\"})"
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  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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    "## Kuramoto Model\n",
    "\n",
    "The Kuramoto model is the classic example that demonstrates phase locking of $N$ coupled oscillators. \n",
    "\n",
    "\\begin{align*}\n",
    "    \\dot{\\theta}_i &= \\omega_i + \\frac{k}{N} \\sum_{j=1}^N \\sin(\\theta_j - \\theta_i) \\\\\n",
    "    \\dot{r}_i      &= 0\n",
    "\\end{align*}\n",
    "\n",
    "The strength of coupling is determined by $k$. "
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   ],
   "source": [
    "k, N, w_m, w_v = 0.2, 10, 10, 0.1\n",
    "KM = GeneralModel(Name=\"Kuramoto\", tstart=0, tend=5, dt=0.01, k=k, N=N, **{\"w_{}\".format(x):np.random.normal(w_m,w_v) for x in range(N)})\n",
    "KM.Initialize(np.append([np.random.uniform(0,2*np.pi) for _ in range(N)],np.ones(N)))\n",
    "KM.Simulate(UpdateMethod=\"RungeKutta\",Flow=\"Kuramoto\")\n",
    "plt.rcParams[\"figure.figsize\"] = (13, 4)\n",
    "KM.PolarPlotState(states={\"theta\":range(N),\"r\":range(N,N+N)},statelabels=[\"x\",\"y\"],legend=[\"osc_{}\".format(x+1) for x in range(N)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
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   "outputs": [
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