{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "%matplotlib inline\n", "%config InlineBackend.figure_format='retina'" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Basics of Dynamics Systems and Bifurcation Theory\n", "# Workshop \\#1, 23 Jan. 2020\n", "\n", "#### Navid C. Constantinou, RSES, ANU, 2020" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Dynamical Systems\n", "\n", "A dynamical system can be expressed in the form\n", "\n", "$$\n", " \\frac{\\mathrm{d}}{\\mathrm{d}t} \\phi(t)= \\mathcal{F}\\big(\\phi(t),\\,t\\big) \\tag{1}.\n", "$$\n", "\n", "Above, $\\phi(t)$ includes all variables needed to describe the state of the system. (Hereafter, dots above variables denote time-differentiation.)\n", "\n", "---\n", "\n", "#### Example: the pendulum\n", "\n", "For a pendulum that evolves freely in three dimensions, $\\phi$ includes the position and the velocity of its free end. In this case, $\\phi$ has six components:\n", "$$\n", " \\phi = [x(t), y(t), z(t), \\dot{x}(t), \\dot{y}(t), \\dot{z}(t)].\n", "$$\n", "\n", "---\n", "\n", "If $\\mathcal{F}$ in Eq. (1) *does not* depend explicitly on time $t$ we say that the dynamical system is **autonomous**. Otherwise, the dynamical system is **non-autonomous**.\n", "\n", "Given equation (1) and the state of the system at some instance $t=t_0$, e.g., $\\phi(t_0)=\\phi_0$, we can predict the state of the system $\\phi(t)$ for all times $t$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## All dynamical systems are of “first order” \n", "\n", "\n", "Any $n$-th order differential equation can be rewritten as a system of $n$ differential equations all being **first order**. In that sense, *all* dynamical systems are first-order in time (but of course their dimensionality may vary).\n", "\n", "\n", "#### Exercise 1\n", "\n", "Show that any **non-autonomous** system of $n$ differential equations corresponds to an **autonomous** system of $n+1$ differential equations.\n", "\n", "*Hint*: You can start by an example: Show that the forced 1D harmonic oscillator\n", "\n", "\\begin{align*}\n", " \\ddot{x} + 2\\gamma\\dot{x}+\\omega^2 x = f_0\\cos(2\\pi t),\n", "\\end{align*}\n", "\n", "can be rewritten as a system of 3 equations that **do not** depend explicitly on time. Then generalise this to what the exercise wants.\n", "\n", "Hint: You may need to change/introduce a new variable." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Exercise 2 (optional)\n", "\n", "Show that any **nonlinear dynamical system** can be rewritten as a **linear** dynamical system of higher dimensionality.\n", "\n", "Take as an example the system:\n", "\n", "\\begin{align*}\n", " \\dot{x} = -x^2.\n", "\\end{align*}\n", "\n", "How big is the dimension of the equivalent linear system for the example above?\n", "\n", "Food for thought: If any nonlinear system can be transformed into a linear one then what's the point of distinguish between *linear* and *nonlinear* dynamical systems?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Attractors of Dynamical Systems\n", "\n", "\n", "**Fixed points** or **equilibrium points** are special values of $\\phi=\\phi^e$ which satisfy:\n", "\n", "$$\n", "\\mathcal{F}(\\phi^e) = 0. \\tag{2}\n", "$$\n", "\n", "They are called equilibria or fixed points because if the system finds itself in one of these states then it will stay there for eternity.\n", "\n", "The character of a fixed point is studied by assuming small deviations about that fixed point,\n", "\n", "$$\n", " \\phi = \\phi^e + \\phi',\\quad\\text{where}\\quad\\|\\phi'\\|\\ll 1. \\tag{3}\n", "$$\n", "\n", "Determining how these small deviations will evolve characterises the fixed point. We do that by writing out the linear dynamical system that governs $\\phi'$. By inserting Eq. (3) in Eq. (1) and after linearization (i.e., discard of the terms that involve quantites which are quadratic or higher order in components of $\\phi'$), we get:\n", "\n", "$$\n", " \\dot{\\phi'} = \\mathcal{F}(\\phi^e + \\phi') \\approx \\mathcal{A}(\\phi^e)\\,\\phi',\\tag{4}\n", "$$\n", "\n", "where $\\mathcal{A}$ is a linear operator that depends on the fixed point $\\phi^e$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "---\n", "\n", "#### Example: the 1D pendulum\n", "\n", "
\n", "\n", "\n", "
\n", "\n", "\n", "Consider a pendulum in one dimension. We learn that the angle $\\theta$ from rest is governed by\n", "\n", "$$\n", " \\ddot{\\theta} + \\frac{g}{\\ell} \\sin{\\theta} = 0. \\tag{5}\n", "$$\n", "\n", "Note that this is a second-order dynamical law. But if we define $\\omega(t) \\equiv \\dot{\\theta}$ and $\\phi=[ \\theta, \\omega ]^\\mathrm{T}$ then we can rewrite the second-order equation of motion as:\n", "\n", "$$\n", " \\begin{bmatrix}\\dot{\\theta}(t)\\\\ \\dot{\\omega}(t) \\end{bmatrix} = \\begin{bmatrix}\\omega(t)\\\\-\\frac{g}{\\ell} \\sin[\\theta(t)]\\end{bmatrix}. \\tag{6}\n", "$$\n", "\n", "What are the pendulum's fixed points?\n", "$$\n", " \\phi^e_1 = \\begin{bmatrix}0\\\\0\\end{bmatrix}\\quad\\textrm{and}\\quad \\phi^e_2 = \\begin{bmatrix}\\pi\\\\0\\end{bmatrix} \\tag{7}\n", "$$\n", "\n", "---" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Asymptotic/modal stability\n", "\n", "\n", "We say that $\\phi^e$ is ***attracting*** or ***asymptotically stable*** (or ***modally stable***) if all trajectories that start near $\\phi^e$ approach it as $t\\to \\infty$.\n", "\n", "In other words, any trajectory $\\phi^e(t)$ that starts within a \"distance\" $\\delta$ of $\\phi^e$ is guaranteed to converge to $\\phi^e$ ***eventually***.\n", "\n", "Asymptotic stability concerns what happens at $t\\to\\infty$. Eigenanalysis of a dynamical system reveals the $t\\to\\infty$ behavior of it." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Recipe\n", "\n", "How do we study the asymptotic/modal stability of these fixed point?\n", "\n", "1. We assume a state that slightly perturbed about the fixed point, $\\phi = \\phi^e + \\phi'$.\n", "2. We insert the state in the equation of motion, use that $\\phi^2$ satisfies (2), and discard any terms which are products of $\\phi'$, e.g. discard $(\\phi')^2$, $(\\phi')^3$, ...\n", "3. We obtain a linear system of the form\n", "$$\n", " \\dot{\\phi'} = \\mathcal{A}\\,\\phi'. \\tag{8}\n", "$$\n", "where operator $\\mathcal{A}$ depends on the equilibrium state $\\phi^e$.\n", "4. Search for eigensolutions $\\phi' = \\widetilde{\\phi}\\,e^{\\sigma t}$. This transforms (6) in an eigenvalue problem with $\\sigma$ the eigenvalue and $\\widetilde{\\phi}$ the eigenvector/eigenfunction:\n", "$$\n", " \\mathcal{A}\\,\\widetilde{\\phi} = \\sigma \\, \\widetilde{\\phi}. \\tag{9}\n", "$$\n", "5. Eigenanalysis of $\\mathcal{A}$ determines its spectrum (spectrum is the set of all eigenvalues).\n", "6. If we have *at least one eigenvalue* $\\sigma$ with $\\mathrm{Re}(\\sigma)>0$ then we have **modal instability**. Otherwise the equilibrium $\\phi^e$ is **modally stable**." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Recipe - continued\n", "\n", "**A bit more on point 6**: We say that if we have an eigenvalue $\\sigma$ with $\\mathrm{Re}(\\sigma)>0$ then we have modal instability. That is because if we have a perturbation which initially looks like the eigenfunction of that fixed point, $\\phi'(t=0) = \\widetilde{\\phi}$, then that perturbation will grow away from $\\phi^e$ at an exponential rate.\n", "\n", "#### Exercise 3\n", "Confirm the above statement and then generalise it. What happens in the above system if initially we start of with any random perturbation (not with the \"special\" perturbation $\\phi'(t=0) = \\widetilde{\\phi}$)? Will the system again move away from $\\phi^e$ at an exponential rate?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "---\n", "#### Example: the pendulum (continued)\n", "For our pendulum example we have that for $\\phi^e_1$\n", "\\begin{align*}\n", " \\mathcal{A}_1 = \\begin{pmatrix}0 & 1 \\\\ -\\frac{g}{\\ell} & 0\\end{pmatrix}\n", "\\end{align*}\n", "while for $\\phi^e_2$:\n", "\\begin{align*}\n", " \\mathcal{A}_2 = \\begin{pmatrix}0 & 1 \\\\ +\\frac{g}{\\ell} & 0\\end{pmatrix}\n", "\\end{align*}\n", "\n", "Eigenanalysis of $\\mathcal{A}_1$ gives that its eigenvalues are $\\sigma = +i\\sqrt{\\frac{g}{\\ell}}, -i\\sqrt{\\frac{g}{\\ell}}$.\n", "\n", "Eigenanalysis of $\\mathcal{A}_2$ gives that its eigenvalues are $\\sigma = +\\sqrt{\\frac{g}{\\ell}}, -\\sqrt{\\frac{g}{\\ell}}$.\n", "\n", "**None** of the eigenvalues of $\\mathcal{A}_1$ have positive real part $\\Rightarrow$ $\\phi^e_1$ is **modally stable**.\n", "\n", "There is exist one eigenvalue of $\\mathcal{A}_2$ with $\\mathrm{Re}(\\sigma)>0$ $\\Rightarrow$ $\\phi^e_2$ is **modally unstable**.\n", "\n", "\n", "---" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Exercise 4\n", "Fill in the missing dots in the pendulum example. First make sure that you can reproduce the above. Then proceed finding the eigenvectors that correspond to those eigenvalues. How would the fastest growing mode look like? \n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## A geometric look on dynamical systems\n", "\n", "A take-away message from the lectures is that actually obtaining analytic solutions for a dynamical system is tedious (sometimes even impossible) and, furthermore, it provides very little insight.\n", "\n", "Instead, one can get a very good idea of how the dynamical system will behave by drawing the so-called phase portrait.\n", "\n", "The phase portrait is constructed by finding attractors (e.g., fixed points or limit cycles) and the corresponding stability characteristic of these attractors. Oftentimes it helps to draw the nullclines (locus of points where each component of $\\dot{\\phi}$ vanishes)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Approximating the solution of a Dynamical System \n", "\n", "In most cases finding analytical solutions in closed form for a dynamical system is hard or impossible. That is, we want a method for constructing approximations of the solution for:\n", "\n", "\\begin{align*}\n", " \\dot{\\phi}(t) &= \\mathcal{F}\\big(\\phi(t)\\,,\\, t\\big), \\tag{10a}\\\\\n", " \\phi(t_0) & =\\phi_0. \\tag{10b}\n", "\\end{align*}\n", "\n", "How do we proceed?\n", "\n", "Let $\\phi(t)$ the solution of Eqs. (10) we want to approximate. Now use a Taylor-series expansion about $t=t_0$:\n", "\n", "\\begin{align*}\n", " \\phi(t) &= \\phi(t_0) + (t-t_0)\\dot{\\phi}\\big|_{t=t_0} + \\tfrac1{2!}(t-t_0)^2\\ddot{\\phi}\\big|_{t=t_0} + \\dots \\\\\n", " &= \\phi(t_0) + (t-t_0) \\mathcal{F}(\\phi_0,t_0) + \\tfrac1{2!}(t-t_0)^2 \\frac{\\mathrm{d}\\mathcal{F}}{\\mathrm{d}t}\\big|_{t=t_0} + \\dots \\tag{11}\\\\\n", "\\end{align*}\n", "\n", "If $t$ is close to $t_0$ then we can neglect terms of order $(t-t_0)^2$ and higher. This gives the so-called Euler approximation." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Forward Euler Time-Stepping Scheme \n", "\n", "Assume a time-discretization with time-step $\\delta$, i.e.,\n", "\n", "$$\n", " t_n = t_0 + n\\,\\delta,\\ n=0,1,\\dots. \\tag{12}\n", "$$\n", "\n", "Let's denote the values of the approximations of $\\phi(t)$ at times above as $\\phi_n \\equiv \\phi(t_0+n\\delta)$. The Forward Euler approximation is then\n", "\n", "$$\n", " \\phi_{n+1} = \\phi_n + \\delta\\,\\mathcal{F}\\big(\\phi_n, t_n\\big). \\tag{13}\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "import numpy as np\n", "from numpy import exp, sin, cos, pi\n", "import matplotlib.pyplot as plt\n", "\n", "import matplotlib.pyplot as plt\n", "from matplotlib import gridspec\n", "\n", "from matplotlib import rc\n", "# rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size':20})\n", "# rc('text', usetex=True)\n", "rc('xtick', labelsize=16) \n", "rc('ytick', labelsize=16) \n", "rc('axes', labelsize=20) # fontsize of the x and y labels" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Example \n", "\n", "Let's try to integrate\n", "\n", "\\begin{align*}\n", " \\dot{x} &= -x + \\sin t, \\\\\n", " x(0) &= x_0.\n", "\\end{align*}\n", "\n", "In this case, we could actually find a closed-form solution of the system:\n", "\n", "$$\n", " x(t) = e^{-t}\\,x_0 + \\frac1{2}(e^{-t} + \\sin t - \\cos t).\n", "$$\n", "\n", "**IMPORTANT**: Always start with an example that you know what to expect!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## **CAN'T STRESS THIS ENOUGH**\n", "\n", "#### **Always** start with an example that you know what to expect!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## So, let's code this up\n", "\n", "\\begin{align*}\n", " \\dot{x} &= -x + \\sin t, \\\\\n", " x(0) &= x_0.\n", "\\end{align*}\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def RHS(x, t):\n", " xdot = -x + sin(t)\n", " return xdot\n", "\n", "def Euler(xn, tn, dt):\n", " xnext = xn + dt*RHS(xn, tn)\n", " return xnext" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "image/png": { "height": 278, "width": 725 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x0 = 2.0 # initial position\n", "\n", "tfin, nsteps = 12.0, 25 # final time, number of time-steps\n", "t = np.linspace(0, tfin, nsteps); dt = t[1]-t[0]\n", "\n", "x_euler = 0*t; x_euler[0] = x0 #initialize array and set initial condition\n", "\n", "for j in np.arange(0, nsteps-1):\n", " xn, tn = x_euler[j], t[j] \n", " x_euler[j+1] = Euler(xn, tn, dt)\n", "\n", "t_an = np.linspace(0, 12.0, 100*nsteps) # a very dense time array\n", "x_an = x0*exp(-t_an) + 0.5*(exp(-t_an) + sin(t_an) - cos(t_an))\n", "\n", "plt.figure(figsize=(12, 4))\n", "plt.plot(t_an, x_an, label=\"truth\", color=\"dodgerblue\")\n", "plt.plot(t, x_euler, '^', label=\"Euler\", color=\"darkgreen\")\n", "plt.legend(fontsize=20); plt.xlabel(r\"$t$\"); plt.ylabel(r\"$x(t)$\");" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Error associated with Forward Euler \n", "\n", "In Forward-Euler timestepping scheme we ignore all terms of $\\mathcal{O}(\\delta^2)$.\n", "\n", "@ every time-step $\\longrightarrow$ error $\\sim \\delta^2$\n", "\n", "$\\dfrac{t_{\\rm final}}{\\delta}$ time-steps $\\longrightarrow$ total error $\\sim \\dfrac{t_{\\rm final}}{\\delta} \\delta^2 \\sim \\delta$\n", "\n", "\n", "#### Good.. But we can do better!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Runge-Kutta 4th order scheme\n", "\n", "The Runge-Kutta 4th order time-stepping scheme is:\n", "$$\n", " \\phi_{n+1} = \\phi_n + \\dfrac{\\delta}{6}\\big(k_1 + 2k_2 + 2k_3 + k4\\big), \\tag{14}\n", "$$\n", "where\n", "\\begin{align}\n", " k_1 & = \\mathcal{F}\\big(\\phi_n\\,,\\, t_n\\big), \\tag{15a}\\\\\n", " k_2 & = \\mathcal{F}\\big(\\phi_n + \\tfrac1{2} k_1 \\delta \\,,\\, t_n + \\tfrac1{2}\\delta\\big), \\tag{15b}\\\\\n", " k_3 & = \\mathcal{F}\\big(\\phi_n + \\tfrac1{2} k_2 \\delta \\,,\\, t_n + \\tfrac1{2}\\delta\\big), \\tag{15c}\\\\\n", " k_4 & = \\mathcal{F}\\big(\\phi_n + k_3 \\delta \\,,\\, t_n + \\delta\\big). \\tag{15d}\\\\\n", "\\end{align}\n", "\n", "RK4 @ every time-step $\\longrightarrow$ error $\\sim \\delta^5$\n", "\n", "total error $\\longrightarrow$ $\\mathcal{O}(\\delta^4)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Let's code this up as well" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def RK4(xn, tn, dt):\n", " k1 = RHS(xn, tn)\n", " k2 = RHS(xn+k1*dt/2, tn+dt/2)\n", " k3 = RHS(xn+k2*dt/2, tn+dt/2)\n", " k4 = RHS(xn+k3*dt , tn+dt)\n", " xnext = xn + (dt/6)*(k1+2*k2+2*k3+k4)\n", " return xnext" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## and compare the two methods" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "image/png": { "height": 278, "width": 725 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x_rk4 = 0*t; x_rk4[0] = x0 #initialize array and set initial condition\n", "\n", "for j in np.arange(0, nsteps-1):\n", " xn, tn = x_rk4[j], t[j] \n", " x_rk4[j+1] = RK4(xn, tn, dt) \n", "\n", "plt.figure(figsize=(12, 4))\n", "plt.plot(t_an, x_an, label=\"truth\", color=\"dodgerblue\")\n", "plt.plot(t, x_euler, '^', label=\"Euler\", color=\"darkgreen\")\n", "plt.plot(t, x_rk4, 's', label=\"RK4\", color=\"salmon\")\n", "plt.legend(fontsize=20); plt.xlabel(r\"$t$\"); plt.ylabel(r\"$x(t)$\");" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## What about a 2-dimensional system\n", "\n", "Consider the following: \n", "\n", "\\begin{align*}\n", "\\dot{x} &= y,\\\\\n", "\\dot{y} &= -x,\\\\\n", "x(0)&=x_0,\\\\\n", "y(0)&=y_0.\n", "\\end{align*}\n", "\n", "(What's do these equations describe?)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def RHS(x, t):\n", " xdot = np.zeros((2,))\n", " xdot[0] = x[1]\n", " xdot[1] = -x[0]\n", " return xdot" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## The simple harmonic oscillator\n", "\n", "The solution is\n", "\n", "\\begin{align*}\n", " x(t) &= x_0 \\cos t + y_0 \\sin t,\\\\\n", " y(t) &= -x_0 \\sin t + y_0 \\cos t.\n", "\\end{align*}" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "x0 = np.zeros((2,))\n", "x0[0], x0[1] = 1.0, 0.0\n", "\n", "tfin, nsteps = 3*pi, 40 # final time, number of time-steps\n", "t = np.linspace(0, tfin, nsteps); dt = t[1]-t[0]\n", "\n", "x_euler = np.zeros((2, len(t))); x_euler[:, 0] = x0\n", "x_rk4 = np.zeros((2, len(t))); x_rk4[:, 0] = x0\n", "\n", "for j in np.arange(0, nsteps-1):\n", " xn, tn = x_euler[:, j], t[j] \n", " x_euler[:, j+1] = Euler(xn, tn, dt)\n", " xn, tn = x_rk4[:, j], t[j] \n", " x_rk4[:, j+1] = RK4(xn, tn, dt) " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "image/png": { "height": 278, "width": 738 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t_an = np.linspace(0, tfin, 100*nsteps) \n", "x_an = np.zeros((2, len(t_an)))\n", "x_an[0, :], x_an[1, :] = cos(t_an), -sin(t_an)\n", "\n", "plt.figure(figsize=(12, 4))\n", "plt.plot(t_an, x_an[0, :], label=\"truth\", color=\"dodgerblue\")\n", "plt.plot(t, x_rk4[0, :], 's', label=\"RK4\", color=\"salmon\")\n", "plt.plot(t, x_euler[0, :], '^', label=\"Euler\", color=\"darkgreen\")\n", "plt.legend(fontsize=20)\n", "plt.xlabel(r\"$t$\"); plt.ylabel(r\"$x(t)$\");" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Let's plot it in phase space $x$-$\\dot{x}$\n", "\n", "What do you expect to see?" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "image/png": { "height": 387, "width": 395 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(6, 6))\n", "x, xdot = x_an[0, :], x_an[1, :]\n", "plt.plot(x, xdot, label=\"truth\", color=\"dodgerblue\")\n", "x, xdot = x_rk4[0, :], x_rk4[1, :]\n", "plt.plot(x, xdot, \"s--\", label=\"RK4\", color=\"salmon\")\n", "x, xdot = x_euler[0, :], x_euler[1, :]\n", "plt.plot(x, xdot, \"^-.\", label=\"Euler\", color=\"darkgreen\")\n", "plt.xlabel(r\"$x$\"); plt.ylabel(r\"$\\dot{x}$\");\n", "plt.axis('square');" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Exercise 5\n", "Why Euler method systematically spirals out?\n", "\n", "#### Exercise 6 \n", "Consider the damped harmonic oscillator:\n", "\n", "$$\n", " \\ddot{x} + 2\\gamma \\dot{x} + x = 0,\\;\\; x(0)=x_0,\\;\\; \\dot{x}(0)=\\upsilon_0.\n", "$$\n", "\n", "- Take $\\gamma=0.1$ and use numerical parameters as above `tfin=3pi` and `nsteps=40`. Integrate using Euler and RK4. What do you conclude about the stability character of the fixed point $x=\\dot{x}=0$ **just** from the results of numerical integrations?\n", "\n", "- How can you make sure that any conclusions you deduce from numerics are actually what the physics imply and not a nuance of the numerics?" ] } ], "metadata": { "celltoolbar": "Slideshow", "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": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 4 }