{ "cells": [ { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from sympy import *\n", "from sympy.core.compatibility import (iterable, is_sequence, ordered,\n", " default_sort_key, range)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "IPython console for SymPy 1.2 (Python 3.6.8-64-bit) (ground types: gmpy)\n", "\n", "These commands were executed:\n", ">>> from __future__ import division\n", ">>> from sympy import *\n", ">>> x, y, z, t = symbols('x y z t')\n", ">>> k, m, n = symbols('k m n', integer=True)\n", ">>> f, g, h = symbols('f g h', cls=Function)\n", ">>> init_printing()\n", "\n", "Documentation can be found at http://docs.sympy.org/1.2/\n", "\n" ] } ], "source": [ "%matplotlib notebook\n", "init_session()" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "def ode_rewrite(ode, fun, x):\n", " \"\"\"\n", " Rewrite an system of ODEs isolating the highest derivatives\n", " \"\"\"\n", " if not iterable(ode):\n", " ode = [ode]\n", " fun = [fun]\n", " orders = [ode_order(eq, f(x)) for eq, f in zip(ode, fun)]\n", " ode_aux = []\n", " old_var = []\n", " diff_order = []\n", " for cont, eq in enumerate(ode):\n", " f = fun[cont]\n", " order = orders[cont]\n", " ode_aux.append(solve(eq, f(x).diff(x, order))[0])\n", " for cont_diff in range(orders[cont]):\n", " old_var.append(diff(f(x), x, cont_diff))\n", " diff_order.append(ode_order(old_var[-1], f(x)))\n", "\n", " return orders, ode_aux, old_var, diff_order" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [], "source": [ "def ode2sys(ode, fun, x):\n", " \"\"\"\n", " Rewrite an system of ODEs of order n as n equations\n", " of order 1\n", " \"\"\"\n", " if not iterable(ode):\n", " ode = [ode]\n", " fun = [fun]\n", " orders, ode, old_var, diff_order = ode_rewrite(ode, fun, x)\n", " u = symbols(\"u:{}\".format(sum(orders)))\n", " #subs = zip(old_var[::-1], u[::-1])\n", " subs = [(a, b) for _, a, b in sorted(zip(diff_order, old_var, u),\n", " key=lambda val: val[0], reverse=True)]\n", " sys = []\n", " cont_var = 1\n", " for cont, eq in enumerate(ode):\n", " sys = sys + list(u[cont_var:cont_var + orders[cont] - 1])\n", " sys.append(eq.subs(subs))\n", " cont_var = cont_var + orders[cont]\n", " return u, sys" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [], "source": [ "def critical_points(ode, fun, x):\n", " \"\"\"Find critical points of an ODE\"\"\"\n", " u, sys = ode2sys(ode, fun, x)\n", " sol = solve(sys, u)\n", " return sol" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [], "source": [ "def classify_critical_points(ode, fun, x):\n", " \"\"\"Classify stability for critical points for 2D systems\"\"\"\n", " u, sys = ode2sys(ode, fun, x)\n", " sol = solve(sys, u)\n", " A = Matrix(sys).jacobian(u)\n", " stab = {xc: None for xc in sol}\n", " for xc in sol:\n", " B = A.subs({u[k]: xc[k] for k in range(len(u))})\n", " trace = B.trace()\n", " det = B.det()\n", " if det < 0:\n", " stab[xc] = \"saddle\"\n", " elif trace == 0:\n", " stab[xc] = \"center\"\n", " elif det > trace**2/4:\n", " if trace > 0:\n", " stab[xc] = \"unstable spiral\"\n", " else:\n", " stab[xc] = \"stable spiral\"\n", " else:\n", " if trace > 0:\n", " stab[xc] = \"unstable node\"\n", " else:\n", " stab[xc] = \"stable node\"\n", " return stab" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [], "source": [ "def plotdf(ode, fun, x, xlim=(-5, 5), ylim=(-5, 5), nx=51, ny=51,\n", " args={None: None}, ax=None):\n", " \"\"\"\n", " \"\"\"\n", " if ax is None:\n", " fig, ax = plt.subplots(1, 1)\n", " \n", " u, sys = ode2sys(ode, fun, x)\n", " u_fun = lambdify(u, sys[0].subs(args), \"numpy\")\n", " v_fun = lambdify(u, sys[1].subs(args), \"numpy\")\n", "\n", " Y, X = np.mgrid[ylim[0]:ylim[1]:ny*1j, xlim[0]:xlim[1]:nx*1j]\n", " U = u_fun(X, Y)\n", " V = v_fun(X, Y)\n", " ax.streamplot(X, Y, U, V)\n", " ax.set_xlim(*xlim)\n", " ax.set_ylim(*ylim)\n", " return ax" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simple pendulum" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [], "source": [ "f = symbols(\"f\", cls=Function)\n", "x = symbols(\"x\")\n", "ode = f(x).diff(x, 2) + sin(f(x))\n", "u, sys = ode2sys(ode, f, x)" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left [ u_{1}, \\quad - \\sin{\\left (u_{0} \\right )}\\right ]$$" ], "text/plain": [ "[u₁, -sin(u₀)]" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sys" ] }, { "cell_type": "code", "execution_count": 69, "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 = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1, 1)\n", "ax = plotdf(ode, f, x, xlim=(-2, 2), ylim=(-2, 2), ax=ax)\n", "markers = {\n", " \"saddle\": \"d\",\n", " \"center\": \"o\",\n", " \"unstable spiral\": \"S\",\n", " \"stable spiral\": \"S\",\n", " \"unstable node\": \"D\",\n", " \"stable node\": \"D\"}\n", "marker_colors = {\n", " \"saddle\": \"#e41a1c\",\n", " \"center\": \"#d9d9d9\",\n", " \"unstable spiral\": \"#e41a1c\",\n", " \"stable spiral\": \"#377eb8\",\n", " \"unstable node\": \"#e41a1c\",\n", " \"stable node\": \"#377eb8\"}\n", "for pt in sol:\n", " pt_x, pt_y = pt\n", " stab = stab_pts[pt]\n", " if pt_x.is_real and pt_y.is_real:\n", " ax.plot(pt_x, pt_y, marker=markers[stab], color=marker_colors[stab],\n", " markersize=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Nonlinear nonhomogeneous second derivative" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left [ u_{1}, \\quad - \\frac{4 u_{0}}{\\epsilon u_{0}^{2} + 1}\\right ]$$" ], "text/plain": [ "⎡ -4⋅u₀ ⎤\n", "⎢u₁, ─────────⎥\n", "⎢ 2 ⎥\n", "⎣ ε⋅u₀ + 1⎦" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ode = f(x).diff(x, 2) + 4*f(x) + eps*f(x)**2*f(x).diff(x, 2)\n", "u, sys = ode2sys(ode, f, x)\n", "sys" ] }, { "cell_type": "code", "execution_count": 81, "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 = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fig, ax = plt.subplots(1, 1)\n", "plotdf(eq, [f, g], t, xlim=(0, 4), ylim=(0, 2.5), ax=ax)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "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": "3.6.8" } }, "nbformat": 4, "nbformat_minor": 1 }