"]}, {"cell_type": "markdown", "metadata": {}, "source": ["\n", "\n\n", "

"]}, {"cell_type": "markdown", "metadata": {}, "source": ["| Equilibrium point | x | y | $\\lambda$ |\n", "|-------------------------|----------|---------|----- |\n", "| Upper LH index-1 saddle | -2.149 | 2.0778 | 1 |\n", "| Upper RH index-1 saddle | 2.149 | 2.0778 | 1 |\n", "| Upper LH index-1 saddle | -2.6862 | 2.0778 | 0.8 |\n", "| Upper RH index-1 saddle | 2.6862 | 2.0778 | 0.8 |\n", "| Upper LH index-1 saddle | -3.5815 | 2.0778 | 0.6 |\n", "| Upper RH index-1 saddle | 3.5815 | 2.0778 | 0.6 |\n", "| Upper LH index-1 saddle | -10.7446 | 2.0778 | 0.2 |\n", "| Upper RH index-1 saddle | 10.7446 | 2.0778 | 0.2 |\n", "\n", "\n\n", "

"]}, {"cell_type": "markdown", "metadata": {}, "source": ["### Lagrangian Descriptors for revealing Phase Space Structures"]}, {"cell_type": "markdown", "metadata": {}, "source": ["In order to reveal the phase space structures that are responsible for the mechanism that allows and prevents dynamical matching, we use in this work the method of Lagrangian descriptors (LDs), see e.g. {% cite mancho2013lagrangian lopesino2017 naik2019a --file caldera2c %}. Lagrangian descriptors is a trajectory-based scalar diagnostic that has been developed in the nonlinear dynamics literature to explore the geometrical template of phase space structures that characterizes qualitatively distinct dynamical behavior. Details on how they are applied for revealing phase space structures in caldera-like PESs are described in {% cite KGW2019 KGW2019a --file caldera2c %}. In this chapter we focus on presenting the results relevant to dynamical matching."]}, {"cell_type": "markdown", "metadata": {}, "source": ["## Implications for Reaction Dynamics\n", "\n\n", "\n", "As we have described in the introduction, the caldera gets its name from the shape of the PES. However, transport across the caldera is a dynamical phenomenon governed by the template of geometrical structures in phase space, and dynamical matching is just one particular type of dynamical phenomenon that we are considering in this chapter. First, we describe the phase space structures that mediate transport into the caldera.\n", "\n", "For a two DoF system, the fixed energy surface is three dimensional. For energies above that of the upper saddles an unstable periodic orbit exists in the energy surface. This is a consequence of the Lyapunov subcenter manifold theorem {% cite moser1976 weinstein1973 rabinowitz1982 --file caldera2c %}. In a fixed energy surface, these periodic orbits have two dimensional stable and unstable manifolds. Trajectories move away from the periodic orbits along the direction of the unstable manifold in forward time. In the upper left panel of Fig. [fig:3](#fig:fig_panel) we show a segment of the unstable manifold of the upper right-hand saddle directed towards the interior of the caldera.\n", "\n", "The region of the central minimum of the caldera may also contain unstable periodic orbits. The stable manifolds of these periodic orbits direct trajectories towards the central minimum. In the upper left panel of Fig. [fig:3](#fig:fig_panel) we show a segment of the stable manifold of an unstable periodic orbit in the region of the central minimum directed away from the central minimum.\n", "\n", "If the stable manifold of a periodic orbit in the central minimum intersects the unstable manifolds of one of the upper saddles we have a mechanism for trajectories to enter the caldera and be directed towards the region of the central minimum. In dynamical systems terminology this is referred to as a heteroclinic connection. This would inhibit dynamical matching, as trajectories entering the caldera would exhibit (temporary) trapping in the region of the central minimum. If the heteroclinic connection breaks, as might occur if a parameter is varied, the mechanism for directing trajectories towards the regions of the central minimum no longer exists, and dynamical matching is possible. Hence, a heteroclinic bifurcation is the critical phase space structure that inhibits or allows dynamical matching, which we now present.\n", "\n", "In order to explore the formation of a heteroclinic intersection between any stable manifold coming from an UPO of the central region of the Caldera and the unstable manifold of the UPO of the upper-right index-1 saddle, as the stretching parameter of the Caldera PES is varied, we probe the phase space structures in the following Poincare surface of section:\n", "\n", "\\begin{equation}\n", "\\mathcal{U}^{+}_{x,p_x} = \\lbrace (x,y,p_x,p_y) \\in \\mathbb{R}^4 \\;|\\; y = 1.88409 \\; ,\\; p_y > 0 \\;,\\; E = 29 \\rbrace\n", "\\label{psos}\n", "\\end{equation}\n", "\n", "In the middle-left panel of Fig. [fig:3](#fig:fig_panel), we observe that there is a critical value of the stretching parameter ($\\lambda=0.778$) for the formation of this heteroclinic connection. For values of the stretching parameter above the critical value there is no heteroclinic connection between any stable manifold coming from an UPO of the central region of the Caldera and the unstable manifold of the UPO of the upper index-1 saddle (see the upper left panel of Fig. [fig:3](#fig:fig_panel)). The non-existence of these heteroclinic connections results in the phenomenon of dynamical matching. In this case, if we integrate an initial condition inside the region of the unstable manifold of UPO of the upper-right index-1 saddle forward and backward in time, we see in the upper right panel of Fig. [fig:3](#fig:fig_panel) that the resulting trajectory comes from the region of the upper-right index-1 saddle and exits the caldera through the region of the opposite lower saddle without any interaction with the central area of the caldera.\n", "\n", "Now, for values of the stretching parameter equal or above the critical value we have the formation of heteroclinic connections between the stable manifold coming from an UPO of the central region of the Caldera and the unstable manifold of the UPO of the upper-right index-1 saddle, (see middle and lower left panels of Fig. [fig:3](#fig:fig_panel)). This heteroclinic connection destroys the dynamical matching mechanism because many trajectories become trapped inside the lobes between the two invariant manifolds. We can see this better if we choose an initial condition inside a lobe, as we illustrate in the middle and lower left panels of Fig. [fig:3](#fig:fig_panel) and integrate it forward and backward. We observe that the resulting trajectory is temporarily trapped in the central area of the caldera before it exits from this area, see the middle and lower right panels of Fig. [fig:3](#fig:fig_panel). "]}, {"cell_type": "markdown", "metadata": {}, "source": ["\n", "\n\n", "

\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["# References\n", "{% bibliography --file caldera2c --cited %}"]}], "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"}, "toc": {"base_numbering": 1, "nav_menu": {}, "number_sections": false, "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": false}}, "nbformat": 4, "nbformat_minor": 4}