" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calculation of the Period of a Periodic Orbit" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section we calculate the period of the periodic orbits. \n", "From \\eqref{eq:hameq} we have $\\dot{q} = \\frac{p}{m}$. Using this expression, and the expression for the level set of the Hamiltonian defining a periodic orbit given in \\eqref{eq:tp1}, we have\n", "\n", "\n", "\\begin{equation}\n", "\\frac{dq}{dt} = \\pm \\sqrt{\\frac{2}{m}} \\sqrt{h-D \\left( 1-e^{-\\alpha q} \\right)^2},\n", "\\label{eq:po1}\n", "\\end{equation}\n", "\n", "\n", "or\n", "\n", "\\begin{equation}\n", "\\frac{dq}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q} \\right)^2}} = \\pm \\sqrt{\\frac{2}{m}} dt.\n", "\\label{eq:po2}\n", "\\end{equation}\n", "\n", "\n", "We denote the period of a periodic orbit corresponding to the level set with energy value $h$ by $T(h)$. We can obtain the period by integrating $dt$ around this level set. Using \\eqref{eq:po2}, this becomes:\n", "\n", "\n", "\n", "\\begin{eqnarray}\n", "T(h) & = & \\sqrt{\\frac{m}{2}} \\int_{q_+}^{q_-} \\frac{dq}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q} \\right)^2}} -\n", "\\sqrt{\\frac{m}{2}} \\int_{q_-}^{q_+} \\frac{dq}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q} \\right)^2}}. \\nonumber \\\\\n", "& = & \\sqrt{2m} \\int_{q_+}^{q_-} \\frac{dq}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q} \\right)^2}}.\n", "\\label{eq:po3}\n", "\\end{eqnarray}\n", "\n", "Computation of this integral is facilitated by the substitution:\n", "\n", "\\begin{equation}\n", "u = e^{-\\alpha q}.\n", "\\label{eq:subs}\n", "\\end{equation}\n", "\n", "\n", "After computing the integral using integral 2.266 in {% cite gradshteyn1980table --file action_angle %} we obtain:\n", "\n", "\\begin{equation}\n", "T(h) = \\frac{\\pi \\sqrt{2m}}{\\alpha \\sqrt{D-h}}\n", "\\label{eq:period}\n", "\\end{equation}\n", "\n", "There are two limits in which this expression can be checked with respect to previously obtained results.\n", "First, we note that for $h=D$ (i.e. the energy of the homoclinic orbit, or ''separatrix'') $T(D) = \\infty$, which is what we expect for the ''period'' of a separatrix. \n", "\n", "Second, we consider $h=0$, which is the energy of the elliptic equilibrium point. In this case we have $T(0) = \\frac{\\pi \\sqrt{2m}}{\\alpha \\sqrt{D}}$, which is $2 \\pi$ divided by the imaginary part of the magnitude of eigenvalue of the Jacobian evaluated at the stable equilibrium point, as we expect. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calculation of position and momentum: $q(t)$ and $p(t)$, $0 < h < D$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section we derive an expression for $q(t)$. Differentiating the expression for $q(t)$ will give the expression for $p(t)$ through the relation $\\dot{q} = \\frac{p}{m}$. Using \\eqref{eq:po2}, we have\n", "\n", "\\begin{equation}\n", "\\int_{q_+}^q \\frac{dq'}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2}} = \\sqrt{\\frac{2}{m}} t.\n", "\\label{eq:traj1}\n", "\\end{equation}\n", "\n", "\n", "Choosing the lower limit if the integral to be $q_+$ is arbitrary, but it is equivalent to the choice of an initial condition. After computing this integral, we obtain:\n", "\n", "\n", "\\begin{equation}\n", "q(t) = \\frac{1}{\\alpha} \\log{\\frac{\\sqrt{Dh} \\cos \\left(\\sqrt{\\frac{2(D-h)}{m}} \\, \\alpha t \\right) + D}{D-h}}.\n", "\\label{eq:q(t)}\n", "\\end{equation}\n", "\n", "\n", "It is straightforward to check that the period of \\eqref{eq:q(t)} is \\eqref{eq:period}." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calculation of the Homoclinic Orbit: $q_0(t)$ and $p_0(t)$, $0 < h < D$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As noted earlier, the homoclinic orbit, corresponding to $h=D$, is given by the level set\n", "\\eqref{eq:homo1}. Hence, the integral expression for the homoclinic orbits is obtained from \\eqref{eq:po2} by setting $h=D$. Computing the integral gives:\n", "\n", "\\begin{equation}\n", "q_0(t) = \\frac{1}{\\alpha} \\log{\\frac{1 + \\frac{2D}{m} \\alpha^2 t^2}{2}}.\n", "\\label{eq:qhom}\n", "\\end{equation}\n", "\n", "\n", "It is a simple matter to check that $\\lim_{t \\rightarrow \\pm \\infty} q(t) = \\infty$. Subsequently we obtain $p_0(t)$ from $\\dot{q}=\\frac{p}{m}$ as\n", "\n", "\\begin{equation}\n", "p_0(t) = \\frac{4mD\\alpha t}{2D\\alpha^2 t^2+m}.\n", "\\label{eq:phom}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calculation of the Action-Angle coordinates: $I$ and $\\theta(t)$, $0 < h < D$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section we compute the action-angle representation of the orbits in the bounded region following {% cite melnikov1963vk wiggins1990introduction mezic1994integrability --file action_angle %}.\n", "\n", "We consider a level set defined by the Hamiltonian \\eqref{eq:levelset}, for $0 < h < D$. i.e. we consider a periodic orbit with period $T(h)$. Choosing an arbitrary reference point on the periodic orbit, the angular displacement of a trajectory starting from this reference position after time $t$ is given by:\n", "\n", "\n", "\\begin{equation}\n", "\\theta = \\frac{2 \\pi}{T(h)} \\int{0}^{t}dt' = \\frac{2 \\pi}{T(h)} \\sqrt{\\frac{m}{2}} \\int_{q_+}^q \\frac{dq'}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2}}.\n", "\\label{eq:angle1}\n", "\\end{equation}\n", "\n", "Using the substitution \\eqref{eq:subs} and integral 2.266 in {% cite gradshteyn1980table --file action_angle %}, we have:\n", "\n", "\\begin{equation}\n", "\\theta = \\frac{\\pi}{T(h)\\alpha} \\sqrt{\\frac{2m}{D-h}} \\left(\\frac{3\\pi}{2} - \\sin^{-1}\\frac{(h-D)e^{\\alpha q}+D}{\\sqrt{Dh}} \\right).\n", "\\label{eq:angle2}\n", "\\end{equation}\n", "\n", "\n", "The action associated with this periodic orbit is the area that it encloses (in phase space) divided by $2 \\pi$:\n", "\n", "\\begin{equation}\n", "I = \\frac{1}{2 \\pi} \\oint_{H=h} p dq.\n", "\\label{eq:action1}\n", "\\end{equation}\n", "\n", "\n", "Recalling \\eqref{eq:po2}\n", "\n", "\\begin{equation}\n", "dq = \\pm \\sqrt{\\frac{2}{m}}\\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2} \\, dt,\n", "\\label{eq:action2}\n", "\\end{equation}\n", "\n", "\n", "we obtain:\n", "\n", "\\begin{equation}\n", "pdq = m \\dot{q} dq = \\pm \\sqrt{2m}\\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2} \\, \\dot{q}dt,\n", "\\label{eq:action3}\n", "\\end{equation}\n", "\n", "\n", "and therefore\n", "\n", "\\begin{equation}\n", "I = \\frac{\\sqrt{2m}}{\\pi} \\int_{q_+}^{q_-}\\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2} \\, dq'.\n", "\\label{eq:action4}\n", "\\end{equation}\n", "\n", "Using the substitution \\eqref{eq:subs} and integral 2.267 in {% cite gradshteyn1980table --file action_angle %}, we have:\n", "\n", "\\begin{equation}\n", "I = \\frac{\\sqrt{2m}}{\\alpha} \\left(\\sqrt{D}-\\sqrt{D-h}\\right).\n", "%checked\n", "\\label{eq:action5}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Expressions for $q(t)$ and $p(t)$, $0 < D < h$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Increasing the total energy $h$ to the value of $D$ and beyond results in unbounded motion. Trajectories retain the turning point $q_-$, while $q_+$ becomes infinite. The expression for $q_-$ is identical to low energies and is obtained from \\eqref{eq:ham} by setting $p=0$ and solving for $q$. Recall from \\eqref{eq:tpneg} that $q_- = - \\frac{1}{\\alpha} \\log \\left(1 + \\sqrt{\\frac{h}{D}} \\right)$.\n", "\n", "For unbounded trajectories it is not possible to define a (finite) period, but we can obtain an expression for $t$ as a function of position. It can be derived by integrating \\eqref{eq:po2} from $q_-$ to an arbitrary position $q$ by using the substitution \\eqref{eq:subs} as follows:\n", "\n", "\\begin{eqnarray}\n", "t(q) & = & \\sqrt{\\frac{m}{2}} \\int_{q}^{q_-} \\frac{dq'}{ \\sqrt{h-D \\left( 1-e^{-\\alpha q'} \\right)^2}}=\n", " -\\frac{1}{\\alpha}\\sqrt{\\frac{m}{2}} \\int_{e^{-\\alpha q}}^{1+\\sqrt{\\frac{h}{D}}} \\frac{du}{u \\sqrt{h-D \\left( 1-u \\right)^2}},\\nonumber \\\\\n", " & = & \\frac{1}{\\alpha}\\sqrt{\\frac{m}{2(h-D)}} \\log{\\left(\\frac{h-D+De^{-\\alpha q}+\\sqrt{(h-D)(h-D(1-e^{-\\alpha q})^2)}}{\\sqrt{hD}e^{-\\alpha q}}\\right)}.\n", "%checked\n", "\\label{eq:tinf}\n", "\\end{eqnarray}\n", "\n", "We obtain an explicit solution $q=q(t)$ by inverting \\eqref{eq:tinf}.\n", "\n", "\\begin{equation}\n", "q(t) = \\frac{1}{\\alpha} \\log{ \\frac{\\sqrt{hD}e^{2\\beta t} -2De^{\\beta t} + \\sqrt{hD}}{2(h-D)e^{\\beta t}} },\n", "%checked\n", "\\label{eq:qinf}\n", "\\end{equation}\n", "where $\\beta=\\alpha\\sqrt{\\frac{2(h-D)}{m}}$. Differentiating \\eqref{eq:qinf} and using the relation $\\dot{q} = \\frac{p}{m}$ yields the expression of $p(t)$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Implications for reaction dynamics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Once explicit solutions are known, it is insightful to consider how they change when the vector field is changed, slightly. This is considered in the field of perturbation theory. The subharmonic and homoclinic Melnikov \n", "{% cite wiggins1990introduction --file action_angle %} are examples of global perturbation methods that consider the effect of perturbations on the entire unperturbed integrable system, such as the Morse oscillator that we have considered. The type of perturbation that we will consider is a time periodic excitation of the Morse oscillator. The subharmonic and homoclinic Melnikov methods for this system have been previously considered in {% cite guo2003dynamical --file action_angle %}, however, with respect to chaotic dynamics, important technical details for this periodically perturbed Morse oscillator were not considered. We will describe these in detail here, as well as consider the nature of the effect of the parameters of the Morse potential, and the periodic excitation, on chaotic dynamics.\n", "\n", "The homoclinic Melnikov function is a well-known and popular method for proving the existence of chaos, in the sense of Smale horseshoes, in time periodically perturbed one degree-of-freedom Hamiltonian systems. The method allows one to prove the existence of transverse homoclinic periodic orbits to a hyperbolic periodic orbit. Such orbits admit a Smale horseshoe construction. There are two issues with the application of the standard homoclinic Melnikov function to the time periodically perturbed Morse oscillator, and both issues arise from the fact that the saddle point at infinity is parabolic, not hyperbolic. One issue is the fact that the standard derivation of the homoclinic Melnikov function uses hyperbolicity of the fixed point to show that certain terms in the Melnikov function vanish. The other issue involves the construction of the Smale horseshoe map for orbits homoclinic to a parabolic point. Both issue are considered for the perturbed Morse oscillator in {% cite beigie1992dynamics --file action_angle %} where it is shown that the growth and decay properties for the parabolic point in the Morse oscillator are sufficient for the standard Melnikov function to be valid as well as the Smale horseshoe map construction to be valid. Hence, the ''Melnikov approach'' is sufficient for determining the existence of chaos for the time-periodically perturbed Morse oscillator. \n", "\n", "We consider the following time-periodic perturbation of the Morse oscillator:\n", "\n", "\\begin{eqnarray}\n", "\\dot{q} = \\frac{\\partial H}{\\partial p} & = & \\frac{p}{m}, \\nonumber \\\\\n", "\\dot{p} = - \\frac{\\partial H}{\\partial q} & = & -2D \\alpha \\left(e^{-\\alpha q} - e^{-2\\alpha q} \\right) +\\varepsilon \\cos{\\omega t},\n", "\\label{eq:perteq}\n", "\\end{eqnarray}\n", "\n", "\n", "where $\\varepsilon>0$ is the magnitude of the perturbation and $\\omega>0$ the frequency of the perturbation. The unperturbed system \\eqref{eq:hameq} is obtained by setting $\\varepsilon=0$. We formulate the homoclinic Melnikov function using expressions \\eqref{eq:qhom}, \\eqref{eq:phom} for $q_0(t)$, $p_0(t)$, the homoclinic orbit of the equilibrium point $(q,p)=(\\infty,0)$, as follows:\n", "\n", "\\begin{equation}\n", " M(t_0,\\phi_0)=\\int\\limits_{-\\infty}^{\\infty} DH(q_0(t),p_0(t))\\cdot(0, \\varepsilon \\cos(\\omega t+\\omega t_0+\\phi_0)) dt,\n", "\\label{eq:Melnikovdef}\n", "\\end{equation}\n", "\n", "\n", "where $DH$ is the gradient of the unperturbed Hamiltonian \\eqref{eq:ham}, $t_0$ defines the point $(q_0(t_0),p_0(t_0))$ at which the Melnikov funtion $M$ is evaluated and $\\phi_0$ is the time at which $M$ is evaluated. We solve \\eqref{eq:Melnikovdef} using integral 3.723 in {% cite gradshteyn1980table --file action_angle %}.\n", "\n", "\\begin{eqnarray}\n", " M(t_0,\\phi_0)&=&\\varepsilon \\int\\limits_{-\\infty}^{\\infty} \\frac{\\partial H}{\\partial p}(q_0(t),p_0(t)) \\cos(\\omega t+\\omega t_0+\\phi_0) dt,\\nonumber \\\\\n", " &=& \\varepsilon \\int\\limits_{-\\infty}^{\\infty} \\frac{p_0(t)}{m} \\left( \\cos(\\omega t)\\cos(\\omega t_0+\\phi_0)-\\sin(\\omega t)\\sin(\\omega t_0+\\phi_0) \\right) dt, \\nonumber\\\\\n", " &=& -\\varepsilon \\frac{2m}{\\alpha} \\sin(\\omega t_0+\\phi_0) \\int\\limits_{-\\infty}^{\\infty} \\frac{t}{t^2+\\frac{m}{2D\\alpha^2}} \\sin(\\omega t) dt, \\nonumber\\\\\n", " &=& -\\varepsilon \\frac{2m}{\\alpha} \\sin(\\omega t_0+\\phi_0) e^{-\\omega\\sqrt{\\frac{m}{2D\\alpha^2}}}.\n", "\\label{eq:Melnikovsol}\n", "\\end{eqnarray}\n", "\n", "\n", "Clearly, this function has ''simple zeros'' indicating the existence of homoclinic orbits, and the associated Smale horseshoe type chaos.\n", "\n", "The magnitude of the Melnikov function is a measure of the intensity of the chaos. \n", "Since the exponent in the exponential term is negative, $$e^{-\\omega\\sqrt{\\frac{m}{2D\\alpha^2}}}\\leq 1.$$ The exponent is decreased by decreasing $\\omega$ and $m$, and increasing $\\alpha$ and $D$. Note that $M$ is monotonically increasing in dissociation energy $D$, yet the homoclinic orbit ceases to exist for $D=\\infty$.\n", "\n", "The dominant term in the Melnikov functions is $$\\frac{2m}{\\alpha}.$$ To reach the highest intensity of chaos in the system, $m$ has to be large and $\\alpha$ small, whereby the mitigating effect of the exponential term can be countered by choosing $D>\\frac{m}{\\alpha^2}$. Then for any $\\omega$ we can find $(t_0,\\phi_0)$, such that $\\sin(\\omega t_0+\\phi_0)=1$ and\n", "\n", "\\begin{equation}\n", " |M(t_0,\\phi_0)|=\\varepsilon \\frac{2m}{\\alpha} e^{-\\omega\\sqrt{\\frac{m}{2D\\alpha^2}}} > \\varepsilon \\frac{2m}{\\alpha}e^{-\\frac{\\omega}{\\sqrt{2}}}.\n", "\\label{eq:Melnikovmag}\n", "\\end{equation}\n", "\n", "\n", "In figure [fig:2](#fig:ld) we show a comparison of different intensities of chaos using Lagrangian descriptors, a method introduced by {% cite madrid2009ld --file action_angle %} and proven to display invariant manifolds in two dimensional area-preserving maps by {% cite lopesino2015cnsns --file action_angle %}." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An animation showing the effect of varying $\\omega$ can be found here\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# References\n", "{% bibliography --file action_angle --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" }, "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 }