{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "

Shannon Entropy

\n", "

\n", "The Shannon or information entropy is a measure of\n", "uncertainty in \n", "communication theory and is useful as an analytic measure of chaos. For \n", "an experiment with $N$ possible outcomes, each with \n", "probability $p_1, p_2,\\ldots ,p_N$ (with $\\sum_{i=1}^N p_i=1$), the Shannon entropy\n", "is\n", "$$\n", "S=-\\sum_{i=1}^N p_i \\,ln \\, p_i.\n", "$$\n", "This is calculated for the logistic map. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "\"\"\" From \"COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\"\n", " by RH Landau, MJ Paez, and CC Bordeianu (deceased).\n", " Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, \n", " C Bordeianu, Univ Bucharest, 2021. \n", " Please respect copyright & acknowledge our work.\"\"\"\n", "\n", "# ShannonEntropy.ipynb: Shannon Entropy for Logistic map\n", "\n", "from IPython.display import IFrame\n", "from numpy import *\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "image/png": 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L6/nf37zIt3/7EmsWTWfDy50smN7Ade9Zy/pj5+XsFD1xSR0nLmnmijcu5+bHXuFzd23h7P/+DSGB719+CnMmMDoyEhrbGL7Y3s28xgamjTNEcPWCRn6/fR8AO9p7iCVTZfbovYfYKG8rhRBLpPiHHz5N85Q6/uX8Y4r620bxyMvQi0iUtJH/gar+JMcmrcDirO+LgDZv+ZnDlj9USEGN4NIXS/Khmx5n48udXPeetZx//IJM9MnOfT1887c7+M3Wdv7u7NdwxRuX5zV0X0R43ylLOX3FLK75+WbOOGp2psOxUDLyxijG8MW93XllOhzO6vmN/OSJXezrHmDLbr8jtnyGvlQa/U1/2Mnzew7xzQ+so2myDYCqVvKJuhHg28AWVf3vETa7E7hSRG4l3Rl7UFV3i8i9wL+KSLO33TnAVUUot1HFvLK/l0de2k9jQ5SmyVG++uA2Hnupgy+95/CkU8tmTeHz7ziu4GMtnTmFb1/22okWGSDTqRpP5jaGqsqL7T2868TxJ87ywyg3t3WxeXcXdZEQy2eXr7OyVBr9C68eYl5jA3+6On8pyyg/+Xj0pwGXAM+IyFPesk8CSwBU9RvA3aRDK7eTDq/8oLeuQ0SuBR739rvG75g13ENV+f6jr/D5X2we0oEqAv954fElzyw4UXwdeyRjuKern+6BREEe/TFZkTdbdnfxmrlTx4zWKSahkBCS0d9WCiGWSFEfre5c7EZ+UTe/I7fWnr2NAh8dYd2NwI0Flc4IDC0dvfzzz57loRfaecPKWXzqLatIppSDvXGap9SVVaYoFF/HHskY+h2xRxZg6KdPjrKwaRLPtXWxua2Ls46eU3hBCyQSChXdo48nU9SV8YFlFIaNjDUKpqWjlzue3MW9m/fw7K4uGqIhrrngGC45dWnFsyUWwqC8kVu68Q19IR49wDFeh2xHT6wiD750Wt/iavSxRKrqZ1cyzNAbBbDnYD9feXAbP3y8hURKOXFJE1edezRvPX4BC5smVbp4BeNLNyNFprzY3k1jQ4TZBQ7nX72gkfs2v5r5XG4iYSm6Rx9LpsoqQRmFYYbeyBtV5Wu/3s5XH9xOSpX3nrKEj5x5JPOnB9e4Z+MbrMQInbHb93Zz5JypBb+trM7y4lfNq4ChD0lJNHrz6KsfM/RG3lz3wDa+/KttnHfcPK46dxWLvXzorjBWHP32vT286ajZBf/+MQvTHbILmyZVJP9LuEQa/fA8+0b1YS1k5MU3H97Bl3+1jQtPWsQX3xWMeTLHS0a6yWEMD/bG2dc9ULA+D7BgegNNk6MV65iOlEKjT6ZoMo++6jFDb4zJrY+9wufv3sJb1szn3x018pAddXO4MdzefggovCMW0oO8rr/4ROY2ViZlb0k0+kSKaNjN68ElzNAbY/KlB7Zy8rIZfOnP1lZsOrpyMFpn7EQjbnxOX1m5yaRLodHHk0pdEWbEMkqLvXMZo9IfT/Jq1wCnr5zlfKeb3xmbS6PfvrebukiIRc3B7ZdIh1eWoDPWom6qHmshY1T2HOwHCHTYZL6ER4mjf7G9h+WzpgT6jSY9YKr4Gn1dJLjnpFYw6aYG2HOwn99uayee1PQw+JTy0r4etr56iNbOPv7z3Ws4aemMnPu2HegDYEENGProKBket+/t5rhF08tdpKISCZtHX6uYoXeUZEq5fWMLdzzZxiMv7UeH3d8N0RDLZk7hpX09PNfWNaKh3+UZ+lrw6DO5boZ1xvbHk7R09vKOE6o7V89YREJS9DTFcRswFQjM0DvKVx/cxnUPbGP5rCn8zZ+s5Nxj5zN9UhRFEYTZ0+rpHkhw/GfvG/Xm33WgDxGYO939yZ1HSmq2o70H1Yl3xFaakmn0jvfduIAZegfZua+H/3noRd66Zj5fvfiEEUdyRjNRJiPrtm0H+pg9tZ76GoisGCm8srWzF4ClM4PbEQvpGbSKqdGnUkoipebRBwBroYDz223tfPqOZ+joiQHpNAWfufM56sIh/vmtq0cdrj/WkH+AtgP9LGx2X7aBkT36gUT6/EzKMeNVkCh2eGXMu27Mo69+zKMPMHc8uYuP/ehpEinl18+38/X3n0hLRx8Pb23n6retZu4Y0+r5Q/5jo9z8bQf6WFWBBFyVYKTOWN8LLnQu2mohHCrugCnf0Neboa96rIUCyrd+u4O/ve0pXrtsBjdffgoAF379j3z6jmc4ZkEjl5y6dMzfEBGiYRnRo1dVdh3oq4mOWBi5MzaeSBvHoI8AjRRZo497bzom3VQ/1kIBQ1X5r/te4HO/2MJ5x83jOx98La9fMYuf/9XpnLJ8Bl39CT739mPz9j6j4dCIGv3+nhgDiRQLphc+4XaQGGm6vXjKDYOW1uhNuqlFTLoJGF96YBtffXA771m3mH9953GZATwzptRx0wdPZn9PjNnT8o+QGS3krpZi6CH9hpOWN4Z79I4Y+tDIb2+FMPimE+zzUguYoQ8Q1z2wla/8aht/tm4R//bO4w5LLhYKybiMPKS9sZE8et/Q10pnLOTusPS94EjApZtih1fGkknAPPogYC0UEG783Utc90A6TfAX3lm8DJKR0MiGvrWzdgZL+aSlrKHGMCNRBNxzjRS5M9aPRqoL+AOwFhjToxeRG4G3AntV9dgc6z8OvC/r91YBs1W1Q0R2AoeAJJBQ1XXFKngtsXNfD1/45fOcvWpu0dMERyMjh9y1Hehncl2Y6ZPKP0lGpUinCRj64PPPTyTAeW4grdEXtTPWOy/m0Vc/+bTQd4H1I61U1f9Q1bWquha4CviNqnZkbfImb70Z+QJQVT59x7PUh0N8/h3HFj2pVjQcynisw2k70MeCpkmBnOi7UCIhOWziEV/XDnJCM/D7Y4qn0ccyHn2wxxfUAmMaelV9GOgYazuPi4FbJlQiYwh3PLWL323fxyfWHzVmXHwhREeRbtoO9tVMR6xPJBQ6rMMyllTqwqHAP/CKrdH7103Qw05rgaK9c4nIZNKe/4+zFitwn4hsFJErxtj/ChHZICIb2tvbi1WsQNPZE+Pau7awdnET7z1l7Lj4QhhNutnVWTsx9D6RcI7O2GQq8B2xUHyNPuPRm3RT9RSzhd4G/H6YbHOaqp4InAt8VETeONLOqnqDqq5T1XWzZxc+AbNLfPHeFzjYF+ffssIoi00klFu66Y8n2d8TY2FTbcTQ+0RzxJq7kqGx2Bp9LOlG2GktUMwWuohhso2qtnn/9wI/BU4u4vGcZvvebm57/BUuOXVpSSeTrguHcnr0tRZD75Mzjj6lTsgTpdLoLQVC9VOUFhKR6cAZwM+ylk0RkWn+Z+Ac4NliHK8W+M97X2BSNMyVZ60o6XGikdw3f9uB9MxStWbocw0giyfc8OiLHkdv0k1gyCe88hbgTGCWiLQCVwNRAFX9hrfZO4D7VLUna9e5wE+9DqwIcLOq/rJ4RXeXp1oO8Mvn9vC3Z69k1tTS5oGPhEJ0JxOHLW+roQlHsomGD++MTaTUKY1eVYvSsRw36SYwjGnoVfXiPLb5LukwzOxlO4DjCy1YraKq/Ps9zzNzSh2Xv2F5yY+Xa4AQQKs/4UgJIn2qmUj48A5LlzR6gJRCMZ5blusmOFgLVRm/3baPP+7Yz5VnrWBqfekzVETDI0k3fcyd1lBzN3GuFAjxZCqTwjjI+B36xdLpY47kAKoFrIWqCFXlv+7fysKmSbz3lCVlOWauKBPwB0vVljcPXhx9jpGxrkg3QNF0estHHxyshaqIx3d28nTLAf7yzCPLNnVfNBzKeGbZ+KNia41I+PDO2Jhj0k2xYunNow8O1kJVxDd/u4PmyVEuPHFR2Y45knSz99AAc6bVnkcfzRFrnki6E14Jo08dOR7iyRThkAQ+NUQtYIa+StjR3s0DW17lklOXMqmufLlDRpJuBhIpGqK1d3mEc8Sau9IZGy62dJNIBT6jZ61grVQlfPt3LxENh7jkdcvKetxIWDITa/gkU0oypTXXEQvpN5zDZ5jSwM8XCyPPoFUocUfedGqB4F+9DtDRE+P2ja2884SF4544ZKLUhUOZqfJ84jUcNpcrqVk8kXIi57r/sCqWRz+QSFFXpr4kY2LU3p1chXz/kZcZSKS4/A1HlP3YuTofByeUqL3LI9f5SKRSRBwIr4wUObwynnTjAVgLBP/qdYA7n27j9BWzWDFnWtmP7Xc+prK8vNr26A9PE5BIKlEHzkVJNHoHzkstYK1UBezt6ufI2VMqcmy/kzFbvonVtEd/eBx9LJki6kBkSbE1+pgjOYBqAWulChNPpujqTzBjSnm1eR+/My1brqjl+OhojqRmzgyYKrJGH0+aRx8UrJUqTGdPDIAZU+sqcnzfmGd3QNa0dJMjqZkr4ZXF1uhjZugDg7VSheno9Qz95MoYet/Ly558ZKCG08/mmoXJFUNfCo3ehfNSC1grVZiObs/QT6mMoa/LJd0ka1mjz2Xo3YgXL7pGn0xZnpuAYK1UYfZ70s3MapJuatqjT0chqQ4aw0Qq5caAqRJo9ObRBwNrpQrT0VNZj96/+bN121rOMz68c1pVPY8++OeiFGmKa/GtL4hYK1UY39A3TYpW5Pi5pJtanjlouNfryxwuhVcWVaOvQWcgiFgrVZiOnhhNk6MVkwb8EZ9DPPpajqP3vV4vlt6fhMQFg+aHiBYz100tXiNBxFqpwnT0xCom28CgAYtb1A2Qnco3bQx9GSvihEdfilw3tXeNBBFrpQqzv2eAmZU09Dmlm/TnWvTWIsM6pxMO9VcUW6O3XDfBYcyrV0RuFJG9IvLsCOvPFJGDIvKU9/eZrHXrReQFEdkuIv9UzIK7QmdPnOYKxdBDVgqEXNKNA8ZtvESHyRv+Q8+lpGaW66b2yKeVvgusH2Ob36rqWu/vGgARCQNfA84FVgMXi8jqiRTWRfb3xCoWWgnZ4ZXZKRCSQG0a+nBo6PnwH4BupEAotkZv4ZVBYcxWUtWHgY4CfvtkYLuq7lDVGHArcEEBv+MsqZTS2VtZjd738mJDUiB4HZAOGLfxkpGyvM7YuEODxyKhwx/qhZJKKYkanZwmiBSrlV4nIk+LyD0icoy3bCHQkrVNq7csJyJyhYhsEJEN7e3tRSpWddPVHyeZ0oolNINBr93i6NMMN4a+9+uCRz+YAmHiGn0tXyNBpBit9ASwVFWPB74K3OEtz3VnjOhKqOoNqrpOVdfNnj27CMWqfgYHS1Umhh5ySze1PvEIDD74XMrkWcwUCLWcJiOITLiVVLVLVbu9z3cDURGZRdqDX5y16SKgbaLHc4lBQ185jz63dJMiGhZEgu/FjpfhHZaZAVMOePT+Q6wYnbG13GEfRCbcSiIyTzyLICIne7+5H3gcWCkiR4hIHXARcOdEj+cSmTw3FdTo/Rs1MSwffa16apnwytTQ8Eo3PHpfppu4oa/l0dNBJDLWBiJyC3AmMEtEWoGrgSiAqn4DuBD4iIgkgD7gIk1nhEqIyJXAvUAYuFFVnytJLQJKpfPcQO4c5bUcNhcNDR1XMDhgKvjno6gafQ3Le0FkTEOvqhePsf564PoR1t0N3F1Y0dynGgx9rpGxtRw2FxnWZ+H/r4s4IN0UUaPPePQ16hAEDWulCtLRE2NyXZiGaLhiZagLH/46X8sefThjDIeGV7rg0YdCQkiKo9HXcod9ELFWqiCVznMDuaWbgRqeIi4zMjY5bGSsA52xkH5gFUOj96Ubm3gkGFgrVZD9VWDowyFB5PCJR2rVU8vE0Ts4YArS7V0MjX5wUJ0b58V1rJUqSGcVGHoRIRoKERs2lWCte/TxzIApPwWCG+cj11SJhWDhlcHCWqmCVIN0A2njdljUjSOGbbwMn0A7nnAnjh7SUl0xNPrB8Eo3zovr1ObdXCVUOkWxTzQSGird1HDUzfBsnn7OG1fOR7hIGn0tz1kQRKyVKkRvLEF/PEVzFRj6yHDppoajboZneEw4pkVHiqbRu9V34TrWShWiowpGxfrUhWWIR1/LMwcNJjUbFl7piERhGn1tYq1UIaohz41PJBw6bMBUrXpqI0084sr5iISkKGmKLXtlsLBWqhD7q2BUrE+6M9aibiBrwNTwiUccmDMW/PBKy3VTa1grVYiO7moy9CGLuvHIdMYOS2oWdsTQR0KhTMjoRHvj+7QAABkcSURBVDDpJlhYK1WIzt7qNfTxpBJ1ILdLIUSGefSxpFIXDjmTsjkSLo5Hb/nog4W1UoXY3xMjGhYaG8bMK1dyosM66NIefeXy71SScGh41E3KmY5YSD/IipkCwaSbYGCtVCE6umM0T66rCk8xEg5lblyo7fBKEfE6LAejblwyZsXU6MMhcUbSch13ruCAUQ15bnzqsqQbVU13xjrkxY6X7BDEeEqdGv1ZTI3eZJvgYC1VITp7q8fQDzFsmfzrtXtpREODD76EYx590TT6Gn7rCyLWUhWiWvLcQFpn9aUbi4/2HnxZaYpd0ujDxdLok+rUA9B1rKUqRDUZ+rpwaNCjt042wqGs8+GaR18kjT6WSFku+gBhLVUBEskUB/viNE+uDkMfycpeaR69F4WU3RnrwOxSPpGsh/pESD8A3XnTcR13ruAAcbAvDkDz5GiFS5ImGg5lPHmb9Hlon0XCMemmWEnNTKMPFmO2lIjcKCJ7ReTZEda/T0Q2eX9/EJHjs9btFJFnROQpEdlQzIIHGX+wVDVkrgQvBULKHyBkHn12Z2zMMekmXKRcN/EaTpMRRPJpqe8C60dZ/xJwhqquAa4Fbhi2/k2qulZV1xVWRPfo7PU9+mox9FmGzTz6IbHmiaRr4ZVFyl7p2APQdcYclqmqD4vIslHW/yHr6yPAookXy206qyihGaQN/fAkXrXsraWzeQ6eD5fORSQcKkpn7IDF0QeKYrfUh4B7sr4rcJ+IbBSRK0bbUUSuEJENIrKhvb29yMWqLnzppqlKNPpIWDKSjQ1t91NC+DNMqTPzxYLv0Rdn4hGXHoCuU7REKyLyJtKG/vSsxaepapuIzAHuF5HnVfXhXPur6g14ss+6desm7nJUMdUm3dTlkm5q+CbOztmecGyUcLE0+lgiRd3k2r1GgkZRWkpE1gDfAi5Q1f3+clVt8/7vBX4KnFyM4wWdzp4YdZEQk+uqI3FYJBRCNT0htnXGDk0TEE+mMrNOuUCxNHrXxhe4zoRbSkSWAD8BLlHVrVnLp4jINP8zcA6QM3Kn1ujsjdE8OVoVCc2ATErieDJlnbEMHRmbSCpRhx56xdLoLbwyWIwp3YjILcCZwCwRaQWuBqIAqvoN4DPATOB/PMOV8CJs5gI/9ZZFgJtV9ZclqEPg6OytnsFSMGjU48mUefSkjWFPLAl40SUOZWhMpykuhkavNX2NBI18om4uHmP95cDlOZbvAI4/fA+jsydWVYben2wjntTBqJsa9uijWWmKE47ldClWmuKBhEk3QcJaqgJ09sZonlIdETdARppIZEk3LskV4yU7w2PcwYlHEilFdWLGPpZIWq6bAGEtVQEOVJl04+dyiZlGD6Q7Y+PZuW4cOhd+qOhEnfq4YwPJXMedKzggpFLqdcZWkaGPDEo3MctHf1h+fpcMWjg02PE+EWIWRx8orKXKzKH+BCmtnsFSMDg4KmEePeCFV/pRN6mUcwOmgAnp9MmUkky51XfhOtZSZaajt7rSHwCZOPEh0k0Ne2v+yFhV9Tx6d87F8MnPC8HSZAQPa6kyk8lcWUXSTZ0n3SS8qJtan/TZHz3qG0OXwiv9h9ZEPPqYRWYFDmupMnOgylIUw6BH78fRu6RJF4KfzdOXb1yKQMp49BPQ6O2tL3hYS5WZjp7qmnQEBr08X7qpdU/ND0H0PdeIQx59pAjSjfXjBA9rqTJzIJO5sno8+mzpJh1NUR05eCpFxEvbnHBQi44UQbrxNXqX+i5cx1qqzHT2xgiHhMaGoiUOnTBDpJuEW9kaC8FP5evnpHctqRkUyaN36AHoOtZSZaajJ15VCc1g0DOLJ9WSVZGOo0/poEFzaWRsUTR68+gDh7VUmTnQG6sq2QbIdL7GkymbUIJBA9YXTyc2c0mLLqZHbykQgoO1VJnp6Ikxo+oM/VDpptY9Nd8Y+obeJY++UI3+p0+28uONrQAZSavWHYIgYS1VZg70xqtqVCxkJzVTG9rOoLzR56UqdunBV6hH/90/vMz1v94O2HSTQcRaqsx09saqalQsDA4IsvDKNL4B64/7ht4dj75Qjf5Qf5yd+3voGUgQS3qSVo07BEHCWqqMqKYTmlWfRp+V68Y8+oxU02sefYauvgSq8PyeQ8QS3kAyhx6AruPOFRwAemJJ4kmtqsFSMGjYMlE3Dhm2QvDTNmc0epfCKwvU6A/1pwf6bdndlYm6sc7Y4GAtVUY6e6ov/QEMHRlrUTdZGr0fdRNxx3MtJE1xfzzJgKfLb97dRTwzMra2B9YFidq+o8tMNSY0g2zpRi3qhsE3nP6Ygx59AWmKD/UnMp+zPfqoQw9A13HnCg4Anb3p198ZVTSNIKS9vJBkjYytcY/ef9D5Gr1L4ZWFpCn2ZZsZU+p4fvehTCd1rUt8QSKvlhKRG0Vkr4g8O8J6EZGviMh2EdkkIidmrbtURLZ5f5cWq+BBxJduqq0zFryMjakUsaTWvKEfHkfvkkErJE1xl+fRn7xsBn3xJNv2dqd/q8avkyCRb0t9F1g/yvpzgZXe3xXA1wFEZAZwNXAKcDJwtYg0F1rYoFOt0g14hj6hxBJJpwxbIWSkm8yAKXfORyEave/Rn7p8BgBPtxwA3HoAuk5eLaWqDwMdo2xyAfB/muYRoElE5gNvBu5X1Q5V7QTuZ/QHhtN09sYRgemTqku6gXSoXNzCK4FBTX5wwJQ70k0hGn1XX9qjP3FpM5GQ8PyeQ4AZ+iBRrJZaCLRkfW/1lo20/DBE5AoR2SAiG9rb24tUrOqisyfG9EnRqpy9KRIOZTI21voN7Hv0fXEH4+jD49fouzyPftbUelbMmUoypURCQqgKr2MjN8W6gnO1uI6y/PCFqjeo6jpVXTd79uwiFau66OyNVaVsA2nvrD+eskmfObwz1qXz4b+tjC/qJm3opzVEWDW/EbBRsUGjWK3VCizO+r4IaBtleU1yoDdedYOlfKJhoTeWfkWv9ZvYlzf6HUxqVkgKhK6+BCGBKXURVs2fBrj18KsFitVadwIf8KJvTgUOqupu4F7gHBFp9jphz/GW1SQdPdXr0UfCIXoGLIcJZGn0DkbdFJIC4VB/nGkNUUIhYfX86YBdI0Ejr2mOROQW4Exgloi0ko6kiQKo6jeAu4HzgO1AL/BBb12HiFwLPO791DWqOlqnrtMc6I2xekFjpYuRk2g4RI/v0TvkwRZCRqPPDJhy53z4dRtveOU0b0Y036N36eFXC+Rl6FX14jHWK/DREdbdCNw4/qK5harS0RurbunGPHpgMMrG9+irsfO8UPy3lfF69I0N6et25tR65jbW1/w1EjSstcpEV1+C/niKOdMaKl2UnGR79LWuv2aHV9aFQ1U17eNEKVSjb5w06BOuWdRUVXMeG2NjrVUmWjp7AVg8Y1KFS5KbaFjoGbDOWBia1MyljlgoTKPv6o+zeMbkzPfPv/3YzNuOEQzM0JeJ1s4+ABY1Tx5jy8qQ9ujd63wshOw5YydF3crQGPLyGo03qZkv3QDMaazOt1JjZGr7ji4jrZ5Hv6i5Wj360OAUcTXu0ftefDqTp1sePaSlKX/e13zo6otnOmONYFLbd3QZae3sY1p9pCrTH8DQyJL6Wvfos9ISu9hfEQ4JyVR+Gn0qpXTHEjRW6XVr5Id7V3GV0trZy8LmSVXbsZftxde8Rp/lxbum0UP6oZ6vRn9oID2FoHW+BpvavqPLSGtnX9Xq8zBUl3fRix0P2W83Lp6LSFjy1ui7+tLpD7I1eiN4uHcVVyGqSktHb9Xq8zDUuNW6R59t3KMOzS7lEx6HRu/PLpUdXmkED/eu4irkQG+cnlhySIhatWHSzSDhkOArbC5OlxcZh0bflUloZh59kKntO7pMDIZWVq9HH8326B2UK8aL/4bj0nyxPpHwODR636M3Qx9o3LuKq5BqD62EoXJFrXv0MGjg3QyvLECjN+km0NgdXQZaMoY+INKNefSZaBsXO2PDISGRt0Zv0o0LuHcVVyGtnX00NlRvDD0MlW5qfcAUDBp4l+aL9YmE0rOJ5YM/MbgNmAo27l3FVUi1h1bCMOnGQeM2Xvx8Ny6mbB5veOWkaNjJN5tawlqvDLR2VndoJQz1XF3UpcdL1OXO2PEMmOpPmD7vAO5dxVVGOoY+CB6978G6lZa3UPwHn4sy1ng0+q6sXPRGcHHvKq4yOnpi9MWTVZue2MePtLGImzSZzliHJh3xGY9GfyhrdikjuNhdXWKqPT2xjy9RmKFPE82EV7p3Psal0ffHLaGZA7h3FVcZLQGIoYdB6cb0+TR+Z6yLSc3C49Do0ymKzdAHHTP0JSYIo2Jh0HM1jz5N1OE4+si44ugTlrnSAfK6ikVkvYi8ICLbReSfcqz/kog85f1tFZEDWeuSWevuLGbhg0BrZy9Nk6NV7xVlDL2Dhq0QMp2xTnr0obw8elU16cYRxnxUi0gY+Brwp0Ar8LiI3Kmqm/1tVPXvsrb/K+CErJ/oU9W1xStysEjH0Fe3Nw9ujwQthEyuGwfPRzScX1KzgUSKeFKtM9YB8rmKTwa2q+oOVY0BtwIXjLL9xcAtxSicC7R09LKoqbo7YmHQk6836QZw+8GXb3il5aJ3h3we1QuBlqzvrcApuTYUkaXAEcCDWYsbRGQDkAC+oKp3jLDvFcAVAEuWLMmjWJUlnkxxz7N76Pcm1B6J1s4+3nTUnDKVqnBMox9KJqmZk+GV+XXGdmVy0ZuhDzr5GPpcV/pIV8lFwO2qmm39lqhqm4gsBx4UkWdU9cXDflD1BuAGgHXr1uU/c3GFeHhrO399y5N5bbt6QWOJSzNxXPZgCyHTGevggy8SDuUVXjmYi96km6CTTwu2Aouzvi8C2kbY9iLgo9kLVLXN+79DRB4ird8fZuiDxs796bDJe/7mDaN6PJGQMLexoVzFKhjz6Ifie/QRZz36sTV6k27cIR9D/ziwUkSOAHaRNubvHb6RiBwFNAN/zFrWDPSq6oCIzAJOA75YjIJXmtbOXqbUhTl63jQnUgZkp0AwBicId/HBl69GPzjpiHn0QWfMFlTVhIhcCdwLhIEbVfU5EbkG2KCqfsjkxcCtqpp9Ba0C/ldEUqQ7fr+QHa0TZPz8NS4YeRj06F2UKgrBkpoNSjem0QefvB7Vqno3cPewZZ8Z9v1fcuz3B+C4CZSvaglCRsrx4Bv6evPogcGwShdHxuar0ds0gu5gd3UBqCq7OvuqerLv8RJ1WKooBJelrPFo9JGQ0BB17xzUGtaCBdDVl+DQQMJJj96ibtI4n+smT42+cVLUGXmylrG7ugCCkqhsPETMox9CxOXslZ5GP7Q77XC6+uMWWukI7l3FZaA1AJN9j5c6C68cgsvZPP3+h7Fk+q4+m3TEFeyuLgA/I+Vihwy9STdDiTh8PnxZaiyd3qYRdAf3ruIy0NrZx7T6iFM3QX0kxNT6CLOn1Ve6KFVBxPHwSmBMnb6zN2YevSO4Y6nKSEtHLwubJznVSRUJh3jg789gxpS6ShelKhicccudNvYZ9OhHNvSqStuBfs4MQJ4mY2zM0BdAq2OhlT7zpld/qoZy4XdOu+jR+3LUaLH0mbmOHQo4qGXcu4pLjKrS2tlb9ZN9GxPD74R1NbwSRtfoWwIy17GRH2box8mB3jg9saTdAI6TkW4c7IzNR6PPRJaZQ+ME7l3FJcbFGHrjcDLSjYOG3vfoR5NuWs2jdwr3ruISE5TJvo2J4XKaYl+jH60ztrWzl+bJUabWWzeeC5ihHycuDpYyDueso+dw5ZtWsLDJvQd6RqNPjqzRp+c6tmvcFczQj5PWzj4aGyJMt9StTjNvegMfe/NRhBz06CN5hFe2dLiVnbXWMUM/TtI3gHk6RnCJjBFemY4s6zND7xBm6MeJ3QBG0BnLo9/XHWMgkXJyrEitYoZ+HPiejt0ARpAZS6Nvtcgy5zBDPw780YJ2AxhBZiyP3gZLuYcZ+nFgscWGC4yl0fsevYsRR7WKU0Gyx3/2PvriyZL9fsq7McyjN4LMWEnNWjv7mDGljikWQ+8MebWkiKwHvgyEgW+p6heGrb8M+A9gl7foelX9lrfuUuDT3vLPqepNRSh3Tj7wuqXE85gibSLMmBLl6HnTSnoMwyglkTE1+j5LZuYYYxp6EQkDXwP+FGgFHheRO1V187BNb1PVK4ftOwO4GlgHKLDR27ezKKUfxj+cc1QpftYwnGJMj76jl1XzG8tZJKPE5KPRnwxsV9UdqhoDbgUuyPP33wzcr6odnnG/H1hfWFENwygGo6UpTqWU1gMWQuwa+Rj6hUBL1vdWb9lw3iUim0TkdhFZPM59EZErRGSDiGxob2/Po1iGYRTCaB79vu4BYomUGXrHyMfQ5xoDPvwK+TmwTFXXAA8Avg6fz77phao3qOo6VV03e/bsPIplGEYhjKbRZ0IrbayIU+Rj6FuBxVnfFwFt2Ruo6n5VHfC+fhM4Kd99DcMoL6N59H5opXXGukU+hv5xYKWIHCEidcBFwJ3ZG4jI/Kyv5wNbvM/3AueISLOINAPneMsMw6gQo2n0/liRhU3m0bvEmFE3qpoQkStJG+gwcKOqPici1wAbVPVO4K9F5HwgAXQAl3n7dojItaQfFgDXqGpHCephGEaejJYCobWzl1lT65hUFy53sYwSklccvareDdw9bNlnsj5fBVw1wr43AjdOoIyGYRSR0VIgWB56N7Ghb4ZRY/jTJH7t1y9y86OvDFn3ckcv56yeW4liGSXEDL1h1BhT6yN8+I3LM/MfZ/OaudN47ylLKlAqo5SYoTeMGkNEuOq8VZUuhlFGLHulYRiG45ihNwzDcBwz9IZhGI5jht4wDMNxzNAbhmE4jhl6wzAMxzFDbxiG4Thm6A3DMBxHVEs7x2ohiEg78HKBu88C9hWxOEGgFusMtVnvWqwz1Ga9x1vnpaqaczKPqjT0E0FENqjqukqXo5zUYp2hNutdi3WG2qx3Mets0o1hGIbjmKE3DMNwHBcN/Q2VLkAFqMU6Q23WuxbrDLVZ76LV2TmN3jAMwxiKix69YRiGkYUZesMwDMcJpKEXkQYReUxEnhaR50Tkszm2uUxE2kXkKe/v8kqUtVjkU2dvuz8Tkc3eNjeXu5zFJs+2/lJWO28VkQOVKGuxyLPOS0Tk1yLypIhsEpHzKlHWYpJnvZeKyK+8Oj8kIosqUdZiIyJhry3vyrGuXkRuE5HtIvKoiCwb9wFUNXB/gABTvc9R4FHg1GHbXAZcX+mylrnOK4EngWbv+5xKl7sc9R62/V8BN1a63GVo6xuAj3ifVwM7K13uMtX7R8Cl3uezgO9VutxFqvvfAzcDd+VY9/+Ab3ifLwJuG+/vB9Kj1zTd3teo9+d0r3Kedf4L4Guq2unts7eMRSwJBbT1xcAtJS9YCcmzzgo0ep+nA21lKl7JyLPeq4FfeZ9/DVxQpuKVDO+t5C3At0bY5ALgJu/z7cCfiIiM5xiBNPSQedV5CtgL3K+qj+bY7F3eK97tIrK4zEUsOnnU+TXAa0Tk9yLyiIisL38pi0+ebY2ILAWOAB4sZ/lKQR51/hfg/SLSCtxN+k0m8ORR76eBd3mf3wFME5GZ5SxjCbgO+ASQGmH9QqAFQFUTwEFgXHUOrKFX1aSqrgUWASeLyLHDNvk5sExV1wAPMPhEDCx51DlCWr45k7Rn+y0RaSpvKYtPHvX2uQi4XVWT5StdacijzhcD31XVRcB5wPdEJLD3s08e9f4YcIaIPAmcAewCEmUuZtEQkbcCe1V142ib5Vg2LgXDhQvjAPAQsH7Y8v2qOuB9/SZwUpmLVjJGqjPQCvxMVeOq+hLwAmnD7wSj1NvnIgIu2wxnlDp/CPiht80fgQbSSbCcYJT7uk1V36mqJwCf8pYdLH8Ji8ZpwPkishO4FThLRL4/bJtWYDGAiERIS3Ud4zlIIA29iMz2PVURmQScDTw/bJv5WV/PB7aUr4TFJ586A3cAb/K2mUVaytlRznIWmzzrjYgcBTQDfyxvCYtPnnV+BfgTb5tVpA19eznLWWzyvK9nZb25XAXcWN5SFhdVvUpVF6nqMtKOyoOq+v5hm90JXOp9vtDbZlwefWTCJa0M84GbRCRM+mH1Q1W9S0SuATao6p3AX4vI+aRf6zpIR+EEmXzqfC9wjohsBpLAx1V1f+WKXBTyqTekpYxbx3sDVCn51PkfgG+KyN+Rfo2/zIG651PvM4F/ExEFHgY+WrHSlpBhdf42aWluO2lbdtG4fy/414ZhGIYxGoGUbgzDMIz8MUNvGIbhOGboDcMwHMcMvWEYhuOYoTcMw3AcM/SGYRiOY4beMAzDcczQG0YeeLnPj/I+zxSRZytdJsPIFzP0hpEfK4Bt3uc1wDMVLIthjAsz9IYxBl76412q6qeRXQNsqmCRDGNcmKE3jLFZy1DDfhJm6I0AYYbeMMbmeNLZIRGRlaRn/DHpxggMZugNY2zWAiEReRr4DOmU15eOvothVA+WvdIwxsBLD3uCqh6qdFkMoxDMozeMURCRaUDKjLwRZMyjNwzDcBzz6A3DMBzHDL1hGIbjmKE3DMNwHDP0hmEYjmOG3jAMw3HM0BuGYTiOGXrDMAzH+f+qlSYB5szePwAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "mumin = 3.5\n", "mumax = 4.0\n", "dmu = 0.005\n", "nbin = 1000\n", "nmax = 100000\n", "prob = zeros( (1000), float) \n", "xx = []\n", "yy = []\n", "\n", "for mu in arange(mumin, mumax, dmu): # mu loop\n", " for j in range(1, nbin): prob[j] = 0\n", " y = 0.5\n", " for n in range(1, nmax + 1):\n", " y = mu*y*(1.0 - y) # Logistic map, Skip transients\n", " if (n > 30000):\n", " ibin = int(y*nbin) + 1\n", " prob[ibin] += 1\n", " entropy = 0.\n", " for ibin in range(1, nbin):\n", " if (prob[ibin]>0):\n", " entropy = entropy - (prob[ibin]/nmax)*math.log10(prob[ibin]/nmax)\n", " yy=yy+[entropy] \n", " xx=xx+[mu]\n", " \n", "plt.plot(xx,yy) \n", "plt.title(\"Entropy vs $\\mu$\")\n", "plt.xlabel('$\\mu$')\n", "plt.show() " ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 1 }