{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Bifurcations and phase transitions" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## How and when is a bifurcating system analogous to a system exhibiting second-order phase transition?" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# import python libraries\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from typing import Tuple \n", "from scipy import optimize" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Supercritical pitchfork. $\\dot{x} = rx - x^3$, $V(x) = x^4/4 - rx^2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r >0$:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r = 0$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 0\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 0')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r < 0$:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = -1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = -1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### There is critical slowing down at bifurcation, $r = 0$: asymptotically the time to reach the steady state $t \\to \\infty$. This is analogous to the divergence (of e.g. susceptibility in magnetic systems) in continuous phase (2nd order) phase transitions in the thermodynamic limit (inifinite system). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Subcritical pitchfork. $\\dot{x} = rx + x^3$, $V(x) = -x^4/4 - rx^2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Were the state at $r = 0$ stable, there would be critical slowing down here. However the state is unstable, so tit is not asymptotically approached. -> not analogous to a 2nd order phase transition." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r > 0$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r = 0$:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 0\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 0')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r < 0$:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = -1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-0.25*np.power(x,4) - r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = -1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Saddle-node. $\\dot{x} = r + x^2$, $V(x) = -x^3/3 - rx$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r > 0$:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = -1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-np.power(x,3)/3 - r*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r = 0$:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 0\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-np.power(x,3)/3 - r*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 0')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r < 0$:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = -1\n", "xa = np.arange(-2, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(-np.power(x,3)/3 - r*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = -1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### So, here the state $x = 0$ is approached agebraically from the left, but not from the right. Dynamically the left half would appear analogous to a continuous phase transition. But a slight deviation to $x > 0$ would cause the system to drift away from this state, so it is not fully analogous to the 2nd order phase transition. Constructing a system where $x$ is resricted to $x \\le 0$ could be considered analogous a 2nd order phase-transition system (2PS) to my understanding, though. So, symmetry is required for an \"unlimited\" system to correspond to 2PS." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Transcritical. $\\dot{x} = rx - x^2$, $V(x) = x^3/3 - (r/2)x^2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r > 0$:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 1\n", "xa = np.arange(-1, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(np.power(x,3)/3 - 0.5*r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r = 0$:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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X+miflsyFQ3py8+ShTD+rL13ap3odU1pAhUNEWq22zsfmfRV8vPswaz89TN6ug+wqOwbA4MwOXDtpIJNH9OL8Qd1IT9HZ3fFOhUNEQuKcY8/h46wvOsLaQKFYv+cwVTU+AHp0SGN8dle+/ZlBfG5EJgN7qAttW+Np4TCzacCDQDLwmHPunnrjLTB+OnAMuM45tybqQUUS0PHqOnYeOMr20kp2lPqfTw4fr6kDIC05iTH9O3P1hGzGDejKudndyOrWThcZbOM8KxxmlgzMB6YCRcAqM1vsnNsU1OwyYFjgMRH4U+BZRFqops5HRVUtB4+eYN+RE+wrr2J/4LHvSBX7K06w/0gV+yuqcM7/GTPo37UdQzI7MmFQdwZnduTs/l0Y1beTdj0lIC+3OCYA25xzOwDMbCEwAwguHDOAp5xzDlhpZl3NrK9zbm/040qic85xotbHiRofJ2rr/MO1Pup8jjqfw+f8z3XO4Tv1XuCzuKAJ/cvTqZXzyTYuaLzPOerqgqbp/vldtXWO6jp/nqraukCuf2Y7Xl1H+fEayqtqKD9eS3lVDUeO13Csuq7Bf1/njBT6dMmgd+cMhvXqyYBu7RnSqwNDMjsyqGcHXXlWTvGycPQHdge9LuL0rYmG2vQHTiscZjYHmAOQnZ0d1qDSdvh8joPHqv1/WZdXUVJxgsPHTq5cayivqj21sj1yvIaKqlqqavwr4upan9fxzygtJYn0lCTSU5Jpl5ZE54xUOmekMqhnBzq3S/G/bpdK54wUunVIo3fnDPp09heLdmkqDNI8XhaOhnaCuha08b/p3AJgAUBubm6DbaTtc85xoLKaHaWVbC89yo7SSvYcPh7YFXOCkooqaupOXzxSk43OGal0aZdKp8CKtV/XdnTOSCEjNZn0lGT/CjnVv1LOCDynJhspSUkkJ0GSGclJRlKSkRwYNgMLLMbBu/1PDp48FnBy3D/f9z+fmmbg+V+GzU7lSE9NIi05iaQkHVuQyPOycBQBA4JeZwHFLWgjCerQ0WrW7j5Mwb5ytpccZceBSraXVFJeVXuqTXpKElnd2tG7cwYTB3Wnd5cMendKP7VLplfnDLq3TyMjNUkHdEWaycvCsQoYZmaDgD3ATOCaem0WA7cEjn9MBI7o+EZiqq71UbC3nLW7D/Pxp4dYu/swnwTOEwDo1SmdIZkd+fK4fgzu2ZEhvToyJLMD/bq001/hImHmWeFwztWa2S3AUvzdcZ9wzuWb2Y2B8Q8DS/B3xd2Gvzvut7zKK9FV53Os3X2IFZtL+WD7ATYWl586xtCrUzrjBnTl6+f7u4CO6d+Zzhk6A1kkWsy5tnc4IDc31+Xl5XkdQ0JUVnmCd7eUsqKwlL9tLeXwsRqSk4xzsrpw3sBujBvQjfHZXenbJUO7lUTCzMxWO+dym9O20S0OM/sD8Kxz7oOwJROpZ3tpJa+uK2ZFYSnriw7jHPTsmM6lI3szeWQmnx2aqesZicSYpnZVbQXuM7O+wF+B55xza6OSStq0w8eqeXX9Xl5cXcTa3Ycxg3EDunLrlOFMHtGLMf0667iESAxrtHA45x4EHjSzgfgPXP/ZzDKA54CFzrktUcoobUBNnY93C0t5cU0RywtKqK7zMbJPJ+ZNH8WMcf3o1TnD64gi0kxnPDjunNsF3Avca2bjgSeAX+A/oC3SpN0Hj/GXDz7hlY/3UHa0mh4d0vjmpIGBG/Z01rEKkTh0xsJhZqnANPxbHZcC7wK/jHAuiXMb9xzhkfd28Pr6YpKTzH/DnvFZXDJCN+wRiXdNHRyfClwNfBH4CFgIzHHOHY1SNokzzjne31bGI+9t529bD9AxPYXrPzuYb31mEH26aFeUSFvR1BbHT4FngdudcwejlEfiUG2djyUb9/HIu9vJLy6nV6d05l42kmsmZuv8CpE2qKmD45OjGUTij3OOpfn7ueeNAj4pO8bgzA7c+7WzuWJ8f11qW6QN0x0ApUU2FB3hrtc38dHOgwzv3ZFHrj2PqaN6qxutSAJQ4ZCQ7DtSxX8vLeSlj4vo3j6NX11xFjPPH0CKDniLJAwVDmmWY9W1LHhvB4+8u4M6n2POxYO5efJQHcMQSUAqHNIk5xyvrt/Lb14vYF95FV88uy9zLxvJgO7tvY4mIh5R4ZBGHag8wR0vb+T/8vcxNqsLf7hmPLk53b2OJSIeU+GQBr2xYS/zXtlIZVUtcy8byfWfHUyyDnyLCCocUs/hY9X8YnE+i9YWc3b/Ltx31TkM793J61giEkNUOOSUtzfvZ+6LGzh4tJpbpwznpslDdHkQETmNCodQUVXDf726iRdWFzGyTyeeuO58zurfxetYIhKjVDgS3LaSCuY8tZpPyo5y8+QhfP/SYTrrW0SapMKRwJbm7+NHz68jIzWJ566fxMTBPbyOJCJxQIUjAfl8jt8v28JDb2/jnKwuPHztefTt0s7rWCISJ1Q4Ekx5VQ23LlzL8s0lXHleFnddcRYZqdo1JSLNp8KRQLbur+CGp1fz6cFj3DVjDN+cNFB34BORkKlwJIil+fu47a9raZeWzLPXT2LCIJ0BLiIt40knfTPrbmZvmdnWwHO3Rto9YWYlZrYx2hnbkkff28ENT69maK+OvPq9i1Q0RKRVvDq7ay6w3Dk3DFgeeN2Qv+C/37m0gHOO/166mV8vKeCLZ/flrzdcoIPgItJqXhWOGcCTgeEngSsaauScew/QbWtbwOdz/GzRRuav2M7VEwbw0NXjdRBcRMLCq2McvZ1zewGcc3vNrFdrJ2hmc4A5ANnZ2a2dXFyrqfPxo+fXsXhdMTdcMpi500bqILiIhE3ECoeZLQP6NDBqXiS+zzm3AFgAkJub6yLxHfHgeHUdNz+7hrc3l/Cf00byH58b4nUkEWljIlY4nHNTGhtnZvvNrG9ga6MvUBKpHImkvKqG7/4lj1W7DvLrr5zFNyYO9DqSiLRBXh3jWAzMDgzPBhZ5lKPNKKs8wTWPrmTNp4d4aOZ4FQ0RiRivCsc9wFQz2wpMDbzGzPqZ2ZKTjczsOeBDYISZFZnZdzxJG+NKKqq48pEP2VZSyaOzc/nSOf28jiQibZgnB8edc2XApQ28XwxMD3p9dTRzxaMjx2qY9fhH7DtSxVPfnqhzNEQk4nSXnjh2rLqWbz+5ih2lR1lwba6KhohEhQpHnKqu9XHj/6zh408P8dDV47hoWE+vI4lIgtC1quJQnc9x6/NreW9LKb/92limndXX60gikkC0xRFnnHPc8cpGXl+/l3nTR3HV+QO8jiQiCUaFI878dmkhz330KTdPHsL1Fw/2Oo6IJCAVjjjy8Lvb+dM72/nGxGxu/8IIr+OISIJS4YgTz330Kfe8sZnLx/blv2acpWtPiYhnVDjiwN+3HuCOVzZyyfBM7r9qHMlJKhoi4h0Vjhi3q+woNz+7hiGZHZj/jXNJS9FPJiLe0loohlWeqOX6p/IAeHRWLh3T1XtaRLynNVGM8vkct/11LdtLj/LktyYwsEcHryOJiADa4ohZv1++lTc37Wfe9FE6K1xEYooKRwx6Y8NeHlq+lX8/L4tvfSbH6zgiIv9ChSPGFOwt50cvrGPcgK786gp1uxWR2KPCEUMOHq3m+qfy6JSRwoJrzyMjNdnrSCIip9HB8RhRU+fj5mfWUFJxgudvuIBenTO8jiQi0iBtccSIu5ds5sMdZdz9lbMZN6Cr13FERBqlwhED3t68nyfe38l1F+bwtfOyvI4jItIkFQ6PlZRXcfsL6xnVtzM/mT7S6zgiImekwuEhn8/xoxfWcay6lodmjiM9RQfDRST2qXB46In3d/K3rQe444ujGda7k9dxRESaRYXDIxv3HOHe/9vM1NG9+cbEbK/jiIg0mwqHB45V1/KDhR/TvUMa935trE7yE5G44knhMLPuZvaWmW0NPHdroM0AM1thZgVmlm9mP/AiayTc9VoBOw4c5f6rxtG9Q5rXcUREQuLVFsdcYLlzbhiwPPC6vlrgR865UcAk4GYzGx3FjBHxfxv38dxHn3LDxUP4zFBdvFBE4o9XhWMG8GRg+EngivoNnHN7nXNrAsMVQAHQP1oBI2HvkePMfWk9Z/fvwm1Th3sdR0SkRbwqHL2dc3vBXyCAXk01NrMcYDzwjybazDGzPDPLKy0tDWfWsKjzOW776zqqa308OHOc7uQnInErYteqMrNlQJ8GRs0LcTodgReBHzrnyhtr55xbACwAyM3NdaF8RzQ89rcdfLijjN/++1gGZ3b0Oo6ISItFrHA456Y0Ns7M9ptZX+fcXjPrC5Q00i4Vf9F4xjn3UoSiRtwnB45y/1tb+MLo3lypS4qISJzzan/JYmB2YHg2sKh+A/P3UX0cKHDO3R/FbGHlnOOnL28gLTmJu3R/DRFpA7wqHPcAU81sKzA18Boz62dmSwJtPgNcC3zezNYGHtO9idtyL6wu4oPtZcydPpLeulS6iLQBntyPwzlXBlzawPvFwPTA8N+BuP7zvLTiBL9+vYAJOd25+nydHS4ibYO69kTQL1/N53h1Hb/56tkkJcV1DRQROUWFI0KWF+zntfV7ueXzQxnaS72oRKTtUOGIgMoTtdzxykaG9+7IjZcM8TqOiEhYqXBEwO+WFrKvvIq7vzpWJ/qJSJujtVqYrd51iCc//IRZkwZy3sDTrt0oIhL3VDjCqLrWx09eWk+fzhn8eJpuAysibZMn3XHbqoff3c6W/ZU8PjuXjumatSLSNmmLI0y2l1byh7e3cfnYvlw6qrfXcUREIkaFI0x+9dom0lOS+MWXxngdRUQkolQ4wuCdwhJWFJby/UuHkdkp3es4IiIRpcLRSjV1Pn71egE5Pdoz+8Icr+OIiEScCkcrPfuPT9lWUslPp4/SORsikhC0pmuFw8eqeWDZFi4c0oOpo3VAXEQSgwpHK/x+2VbKj9fws8tH6z4bIpIwVDhaaFtJBU+v3MXMCdmM6tvZ6zgiIlGjwtFCv369gPapydw2dbjXUUREokqFowVOdr/93qVD6dlR3W9FJLGocISoNqj77XUXDvI6johI1KlwhOgZdb8VkQSnNV8I1P1WRESFIyQPLlf3WxERFY5m2nngKE9/qO63IiIqHM304LItpCYn8cMpw7yOIiLiKU8Kh5l1N7O3zGxr4Pm0e6yaWYaZfWRm68ws38x+6UVWgC37K1i0rphZFw6kV6cMr2KIiMQEr7Y45gLLnXPDgOWB1/WdAD7vnDsHGAdMM7NJ0Yv4T79ftoUOaSncePEQL75eRCSmeFU4ZgBPBoafBK6o38D5VQZepgYeLirpguQXH2HJhn18+6JBdOuQFu2vFxGJOV4Vjt7Oub0AgedeDTUys2QzWwuUAG855/7R2ATNbI6Z5ZlZXmlpadiCPvDWFjpnpPCdi3Syn4gIRLBwmNkyM9vYwGNGc6fhnKtzzo0DsoAJZnZWE20XOOdynXO5mZmZYfgXwMefHmJZQQk3XDKELu1SwzJNEZF4lxKpCTvnpjQ2zsz2m1lf59xeM+uLf4uiqWkdNrN3gGnAxvAmbdz9b22he4c0rtOd/URETvFqV9ViYHZgeDawqH4DM8s0s66B4XbAFGBztAL+Y0cZf9t6gP+4ZAgd0iNWX0VE4o5XheMeYKqZbQWmBl5jZv3MbEmgTV9ghZmtB1bhP8bxWjTCOee4760t9OqUzjcnDYzGV4qIxA1P/pR2zpUBlzbwfjEwPTC8Hhgf5WgAvL+tjI92HuSXXx5Du7RkLyKIiMQsnTlej3OO371ZSL8uGcycMMDrOCIiMUeFo54VhSWs3X2Y7186jPQUbW2IiNSnwhHE53Pc9+YWsru352vnZXkdR0QkJqlwBFmav4/84nJ+OGUYqcmaNSIiDdHaMaDO53hg2RaGZHZgxrj+XscREYlZOkEh4HhNHeMHdONzIzJJTtJNmkREGqPCEdAxPYV7/32s1zFERGKedlWJiEhIVDhERCQkKhwiIhISFQ4REQmJCoeIiIREhUNEREKiwiEiIiFR4RARkZCYc87rDGFnZqXArhZ+vCdwIIxxwknZWkbZWkbZWiZesw10zmU2ZyJtsnC0hpnlOedyvc7REGVrGWVrGWVrmUTIpl1VIiISEhUOEREJiQrH6RZ4HaAJytYyytYyytYybT6bjnGIiEhItMUhIiIhUeEQEZGQJGThMLMrzSzfzHxm1mjXNDObZmaFZrbNzOYGvd/dzN4ys62B525hzHbGaZvZCDNbG/QoN7MfBsbdaWZ7gsZNj2a2QLtPzGxD4PvzQv18pLKZ2QAzW2FmBYHf/wdB48I63xpbdoLGm5k9FBi/3szObe5nW6sZ2b4RyLTezD4ws3OCxjX420Yx2+fM7EjQ7/Tz5n42Ctl+HJRro5nVmVn3wLhIz7cnzKzEzDY2Mj68y5tzLuEewChgBPAOkNtIm2RgOzAYSAPWAaMD434LzA0MzwXuDWO2kKYdyLkP/8k7AHcCt0dovjUrG/AJ0LO1/7ZwZwP6AucGhjsBW4J+07DNt6aWnaA204E3AAMmAf9o7mejkO1CoFtg+LKT2Zr6baOY7XPAay35bKSz1Wv/JeDtaMy3wPQvBs4FNjYyPqzLW0JucTjnCpxzhWdoNgHY5pzb4ZyrBhYCMwLjZgBPBoafBK4IY7xQp30psN0519Iz5UPR2n+3p/PNObfXObcmMFwBFAD9w5jhpKaWneC8Tzm/lUBXM+vbzM9GNJtz7gPn3KHAy5VAVhi/v1XZIvTZSEz/auC5MH5/k5xz7wEHm2gS1uUtIQtHM/UHdge9LuKfK5nezrm94F8ZAb3C+L2hTnsmpy+gtwQ2R58I5+6gELI54E0zW21mc1rw+UhmA8DMcoDxwD+C3g7XfGtq2TlTm+Z8tjVCnf538P+lelJjv200s11gZuvM7A0zGxPiZyOdDTNrD0wDXgx6O5LzrTnCurylhDVaDDGzZUCfBkbNc84tas4kGngvLH2Xm8oW4nTSgC8DPwl6+0/AXfiz3gXcB3w7ytk+45wrNrNewFtmtjnwF1GrhHG+dcT/n/qHzrnywNutmm/1v6KB9+ovO421idhyd4bvPb2h2WT8heOioLcj8tuGkG0N/t2ylYHjUK8Aw5r52UhnO+lLwPvOueAtgEjOt+YI6/LWZguHc25KKydRBAwIep0FFAeG95tZX+fc3sDmXkm4splZKNO+DFjjnNsfNO1Tw2b2KPBatLM554oDzyVm9jL+zeH3iIH5Zmap+IvGM865l4Km3ar5Vk9Ty86Z2qQ147Ot0ZxsmNlY4DHgMudc2cn3m/hto5ItqNDjnFtiZn80s57N+WykswU5bS9AhOdbc4R1edOuqsatAoaZ2aDAX/YzgcWBcYuB2YHh2UBztmCaK5Rpn7YfNbDSPOkrQIO9LCKVzcw6mFmnk8PAF4IyeDrfzMyAx4EC59z99caFc741tewE550V6O0yCTgS2MXWnM+2xhmnb2bZwEvAtc65LUHvN/XbRitbn8DviJlNwL8OK2vOZyOdLZCpC3AJQctfFOZbc4R3eYvUUf5YfuBfMRQBJ4D9wNLA+/2AJUHtpuPvebMd/y6uk+/3AJYDWwPP3cOYrcFpN5CtPf7/MF3qff5pYAOwPrAA9I1mNvy9M9YFHvmxNN/w73JxgXmzNvCYHon51tCyA9wI3BgYNmB+YPwGgnr3NbbchXFenSnbY8ChoHmUd6bfNorZbgl89zr8B+4vjJX5Fnh9HbCw3ueiMd+eA/YCNfjXbd+J5PKmS46IiEhItKtKRERCosIhIiIhUeEQEZGQqHCIiEhIVDhERCQkKhwiIhISFQ4REQmJCodIhJnZ+YGLJ2YEziLON7OzvM4l0lI6AVAkCszsV0AG0A4ocs7d7XEkkRZT4RCJgsB1gFYBVfgvk1HncSSRFtOuKpHo6A50xH/nwQyPs4i0irY4RKLAzBbjv7vaIPwXULzF40giLdZm78chEivMbBZQ65x71sySgQ/M7PPOube9zibSEtriEBGRkOgYh4iIhESFQ0REQqLCISIiIVHhEBGRkKhwiIhISFQ4REQkJCocIiISkv8P2H+/lXUPcaAAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 0\n", "xa = np.arange(-1, 1, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(np.power(x,3)/3 - 0.5*r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 0')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $r > 1$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'V')" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "g = []\n", "r = 1\n", "xa = np.arange(-1, 2, 0.05)\n", "def funcs():\n", " for x in xa:\n", " g.append(np.power(x,3)/3 - 0.5*r*x*x)\n", "funcs()\n", "fig = plt.figure()\n", "ax = fig.add_subplot(1, 1, 1)\n", "ax.plot(xa, g, label='r = 1')\n", "ax.legend(loc='best')\n", "ax.set_xlabel('x')\n", "ax.set_ylabel('V')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### As in the saddle-node case and for the same reasons, the whole system is not analogous to the 2nd order phase transition system, but a system restricted to $x < 0$ might be considered as such." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.9" } }, "nbformat": 4, "nbformat_minor": 4 }