{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# One Degree-of-Freedom (DoF) Hamiltonian Bifurcation of Equilibria" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(ADD INTRODUCTORY LANGUAGE ABOUT PROBLEM DEVELOPMENT)\n", "We will now consider two examples of bifurcation of equilibria in two dimensional Hamiltonian system; in particular, the Hamiltonian saddle-node and Hamiltonian pitchfork bifurcations. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hamiltonian saddle-node bifurcation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We consider the Hamiltonian:\n", "\n", "\\begin{equation}\n", "H (q, p) = \\frac{p^2}{2} - \\lambda q + \\frac{q^3}{3}, \\quad (q, p) \\in \\mathbb{R}^2.\n", "\\label{eq:hamApp13}\n", "\\end{equation}\n", "\n", "\n", "where $\\lambda$ is considered to be a parameter that can be varied. From this Hamiltonian, we derive Hamilton's equations:\n", "\n", "\\begin{eqnarray}\n", "\\dot{q} & = & \\frac{\\partial H}{\\partial p} = p, \\nonumber \\\\\n", "\\dot{p} & = & -\\frac{\\partial H}{\\partial q} =\\lambda - q^2.\n", "\\label{eq:hamApp14}\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Revealing the Phase Space Structures and their implications for Reaction Dynamics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The fixed points for \\eqref{eq:hamApp14} are:\n", "\n", "\\begin{equation}\n", "(q, p) = (\\pm\\sqrt{\\lambda}, 0),\n", "\\end{equation}\n", "\n", "\n", "from which it follows that there are no fixed points for $\\lambda <0$, one fixed point for $\\lambda =0$, and two fixed points for $\\lambda >0$. This is the scenario for a saddle-node bifurcation. \n", "\n", "Next we examine the stability of the fixed points. The Jacobian of \\eqref{eq:hamApp14} is given by:\n", "\n", "\\begin{equation}\n", "J =\\left(\n", "\\begin{array}{cc} \n", "0 & 1\\\\\n", "-2 q & 0\n", "\\end{array}\n", "\\right).\n", "\\label{eq:hamApp15}\n", "\\end{equation}\n", "\n", "\n", "The eigenvalues of this matrix are:\n", "\n", "\\begin{equation}\n", "\\Lambda_{1, 2} = \\pm \\sqrt{-2q}.\n", "\\end{equation}\n", "\n", "\n", "Hence $(q, p) = (-\\sqrt{\\lambda}, 0)$ is a saddle, $(q, p) = (\\sqrt{\\lambda}, 0)$ is a center, and $(q, p) = (0, 0)$ has two zero eigenvalues. The phase portraits are shown in Fig. [fig:1](#fig:appC_fig3)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "<img width=\"560\" height=\"315\" src=\"figures/ham_sn.png\">\n", "\n", "<a id=\"fig:appC_fig3\"></a>\n", "<figcaption style=\"text-align:center;font-size:14px\"><b>fig:1 </b><em> The phase portraits for the Hamiltonian saddle-node bifurcation.</em></figcaption><hr>" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hamiltonian pitchfork bifurcation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We consider the Hamiltonian:\n", "\n", "\\begin{equation}\n", "H (q, p) = \\frac{p^2}{2} - \\lambda \\frac{q^2}{2} + \\frac{q^4}{4},\n", "\\label{eq:hamApp16}\n", "\\end{equation}\n", "\n", "\n", "where $\\lambda$ is considered to be a parameter that can be varied. From this Hamiltonian, we derive Hamilton's equations:\n", "\n", "\\begin{eqnarray}\n", "\\dot{q} & = & \\frac{\\partial H}{\\partial p} = p, \\nonumber \\\\\n", "\\dot{p} & = & -\\frac{\\partial H}{\\partial q} =\\lambda q - q^3.\n", "\\label{eq:hamApp17}\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Revealing the Phase Space Structures and their implications for Reaction Dynamics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The fixed points for \\eqref{eq:hamApp17} are:\n", "\n", "\\begin{equation}\n", "(q, p) = (0, 0), \\, (\\pm\\sqrt{\\lambda}, 0),\n", "\\end{equation}\n", "\n", "\n", "from which it follows that there is one fixed point for $\\lambda \\leq 0$, and three fixed points for $\\lambda >0$. This is the scenario for a pitchfork bifurcation.\n", "\n", "Next we examine the stability of the fixed points. The Jacobian of \\eqref{eq:hamApp17} is given by:\n", "\n", "\\begin{equation}\n", "J = \\left(\n", "\\begin{array}{cc} \n", "0 & 1\\\\\n", "\\lambda-3q^2 & 0\n", "\\end{array}\n", "\\right).\n", "\\label{eq:hamApp18}\n", "\\end{equation}\n", "\n", "\n", "The eigenvalues of this matrix are:\n", "\n", "\\begin{equation}\n", "\\Lambda_{1, 2} = \\pm \\sqrt{\\lambda - 3q^2 }.\n", "\\end{equation}\n", "\n", "\n", "Hence $(q, p) = (0, 0)$ is a center for $\\lambda <0$, a saddle for $\\lambda >0$ and has two zero eigenvalues for $\\lambda =0$. The fixed points $(q, p) = (\\sqrt{\\lambda}, 0)$ are centers for $\\lambda >0$. The phase portraits are shown in Fig. [fig:2](#fig:appC_fig4)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "<img width=\"560\" height=\"315\" src=\"figures/ham_pitch.png\">\n", "\n", "<a id=\"fig:appC_fig4\"></a>\n", "<figcaption style=\"text-align:center;font-size:14px\"><b>fig:2 </b><em> The phase portraits for the Hamiltonian pitchfork bifurcation.</em></figcaption><hr>" ] } ], "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": "3.7.1" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "/Users/bas/research/book_sprint/champs-booksprint-planning/convert_tex2html/article_test/sample.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 4 }