{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "import numpy as np\n", "from scipy.integrate import odeint, quad\n", "from scipy.optimize import brentq\n", "import matplotlib.pyplot as plt\n", "from matplotlib import animation, rc\n", "import seaborn as sbs\n", "#rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 20})\n", "#rc('text', usetex=True)\n", "#rc('animation', html='jshtml')\n", "#plt.rcParams[\"animation.html\"] = \"jshtml\"" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def deriv(X, t, omega, gamma, alpha, delta, beta):\n", " \"\"\"Return the derivatives dx/dt and d2x/dt2.\"\"\"\n", " \n", " #dVdx = lambda x: x**3 - x\n", " x, xdot = X\n", " xdotdot = gamma*np.cos(omega*t) - delta*xdot + alpha*x - beta*x*x*x\n", " return xdot, xdotdot\n", "\n", "def solve_duffing(tmax, dt_per_period, t_trans, x0, v0, params):\n", " \"\"\"Solve the Duffing equation for parameters gamma, delta, omega.\n", " \n", " Find the numerical solution to the Duffing equation using a suitable\n", " time grid: tmax is the maximum time (s) to integrate to; t_trans is\n", " the initial time period of transient behaviour until the solution\n", " settles down (if it does) to some kind of periodic motion (these data\n", " points are dropped) and dt_per_period is the number of time samples\n", " (of duration dt) to include per period of the driving motion (frequency\n", " omega).\n", " \n", " Returns the time grid, t (after t_trans), position, x, and velocity,\n", " xdot, dt, and step, the number of array points per period of the driving\n", " motion.\n", " \n", " \"\"\"\n", " # Time point spacings and the time grid\n", "\n", " omega, gamma, alpha, delta, beta = params\n", " period = 2*np.pi/omega\n", " dt = 2*np.pi/omega / dt_per_period\n", " step = int(period / dt)\n", " t = np.arange(0, tmax, dt)\n", " # Initial conditions: x, xdot\n", " X0 = [x0, v0]\n", " X = odeint(deriv, X0, t, args=params)\n", " idx = int(t_trans / dt)\n", " return t[idx:], X[idx:], dt, step" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Set up the motion for a oscillator with initial position\n", "# x0 and initially at rest.\n", "x0, v0 = 1, 0.1\n", "tmax, t_trans = 10000, 100\n", "dt_per_period = 500\n", "\n", "omega = 1.0\n", "gamma = 0.4\n", "alpha = 1.0\n", "delta = 0.1\n", "beta = 1.0\n", "\n", "omega = 1.4\n", "gamma, delta = 0.4, 0.1\n", "\n", "\n", "params = (omega, gamma, alpha, delta, beta)\n", "xs = []" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(1102945,)\n", "(1102945,)\n" ] } ], "source": [ "# Solve the equation of motion.\n", "params = (omega, gamma, alpha, delta, 0.01)\n", "t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0, params)\n", "x, xdot = X.T\n", "xs.append(x)\n", "print(t.shape)\n", "print(x.shape)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# Solve the equation of motion.\n", "#params = (omega, gamma, 1.2, delta, 0.1)\n", "#t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0, params)\n", "#x, xdot = X.T\n", "#xs.append(x)\n", "#print(t.shape)\n", "#print(x.shape)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(1102945,)\n", "(1102945,)\n" ] } ], "source": [ "# Solve the equation of motion.\n", "params = (omega, gamma, alpha, delta, 1.0)\n", "t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0, params)\n", "x, xdot = X.T\n", "xs.append(x)\n", "print(t.shape)\n", "print(x.shape)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def tsanim(tss):\n", " \"Plot an array of time series data t, x = [ti], [xi].\"\n", " nrows = len(tss)\n", " fig, axes = plt.subplots(nrows=nrows, ncols=1)\n", " tsmax = lambda ts: max(len(ts[0]), len(ts[1]))\n", " ts_maxlen = max(map(tsmax, tss))\n", " tinit = int(0.1*ts_maxlen)\n", " print(\"Animate time series from t=%d with %d frames\" % (tinit, ts_maxlen))\n", "\n", " def tsplot(ax, ts):\n", " t, x = ts\n", " ax.set_ylabel(r'$x [\\mathrm{m}]$')\n", " ax.set_ylim(np.min(x), np.max(x))\n", " line, = ax.plot(t[:tinit], x[:tinit])\n", " return line\n", " \n", " lines = list(map(lambda axts: tsplot(*axts), zip(axes, tss)))\n", " axes[-1].set_xlabel(r'$t [\\mathrm{s}]$')\n", "\n", " def animate(i):\n", " \"\"\"Update the image for iteration i of the Matplotlib animation.\"\"\"\n", " for ax, line, ts in zip(axes, lines, tss):\n", " t, x = ts\n", " line.set_data(t[:i+1], x[:i+1])\n", " ax.set_xlim(t_trans, t[i])\n", " return\n", "\n", " #plt.tight_layout()\n", " \n", " #tmp\n", " fig.suptitle(\"Duffing oscillator, beta = 0.01 (top) and 1.0 (bottom)\")\n", " \n", " anim = animation.FuncAnimation(fig, animate, frames=20000, interval=1)\n", " return anim" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('