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"# One Degree-of-Freedom (DoF) Hamiltonian Bifurcation of Equilibria"
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"(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. "
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"## Hamiltonian saddle-node bifurcation"
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"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}"
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"### Revealing the Phase Space Structures and their implications for Reaction Dynamics"
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"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)."
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"<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>"
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"source": [
"## Hamiltonian pitchfork bifurcation"
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"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}"
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"metadata": {},
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"### Revealing the Phase Space Structures and their implications for Reaction Dynamics"
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"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)."
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"<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>"
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