dynamics.ipynb

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    "# Behavioral dynamics\n",
    "\n",
    "Warren, W. H. (2006). The dynamics of perception and action. *Psychological Review, 113*(2), 358-389. http://mlrg.cs.brown.edu/warren-intro.pdf\n",
    "\n",
    "## Theoretical motivation\n",
    "\n",
    "#### The big-picture setup\n",
    "\n",
    "> The **organization of behavior** has been a central concern for psychology for well over a century. How is it that humans and other animals can generate behavioral patterns that are tightly coordinated wit the environment, in the service of achieving a specific goal?\n",
    "\n",
    "> ...organization in behavior has been attributed to prior organization in\n",
    "> - the structure of the nervous system (the neuroreductionist view),\n",
    "> - the structure of internal representations (the cognitivist view), or\n",
    "> - in the contingencies presented by the environemnt (the behaviorist view).\n",
    "\n",
    "> This...merely displaces the original problem of bheavioral organization to preexisting internal or external structures, begging the question of why that particular organization obtains and how that specific structure originated.\n",
    "\n",
    "> Gibson (1979):\n",
    ">> Locomotion and manipulation . . . are controlled not by the brain but\n",
    ">> by information. . . . Control lies in the animal–environment system....\n",
    ">> The rules that govern behavior are not like laws enforced by an\n",
    ">> authority or decisions made by a commander; behavior is regular\n",
    ">> without being regulated. The question is how this can be.\n",
    "\n",
    "> Although the quotation asserts Gibson’s belief that behavior is\n",
    "regular without being centrally controlled, the question of how this\n",
    "can be remains open.\n",
    "\n",
    "> ...**the organization of behavior can be attributed to the dynamics\n",
    "of the agent–environment interaction** under such constraints and\n",
    "seek to develop a coherent theoretical framework for understanding\n",
    "perception and action.\n",
    "\n",
    "#### Stability and flexibility\n",
    "\n",
    "> Human and animal behavior exhibits two complementary attributes\n",
    "that need to be accounted for: *stability* and *flexibility*.\n",
    "\n",
    "> ...behavior is characterized by stable and reproducible\n",
    "low-dimensional patterns. These patterns are stable in the sense\n",
    "that the functional form of movement is consistent over time and\n",
    "resists perturbation and reproducible in that a similar pattern may\n",
    "recur on separate occasions.\n",
    "\n",
    "> ...behavior is not\n",
    "stereotyped and rigid but flexible and adaptive. Although action\n",
    "patterns exhibit regular morphologies, the agent is not locked into\n",
    "to a rigidly stable solution but can modulate the behavioral pattern.\n",
    "To the extent that such flexibility is tailored to current environmental\n",
    "conditions or task demands, it implicates perceptual\n",
    "control."
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    "## What are dynamical systems?\n",
    "\n",
    "A dynamical system is an abstract mathematical object built around the general concept of *change over time*.\n",
    "\n",
    "At any time, a dynamical system has a state: one or more values that completely characterize the system. The system's dynamics describe how the state changes, as a function of the state.\n",
    "\n",
    "$$\n",
    "\\dot{\\mathbf{x}} = \\mathbf{f}(\\mathbf{x})\n",
    "$$\n",
    "\n",
    "where $\\mathbf{x}$ is the state (bold means this is a vector) and $\\mathbf{f}$ is the dynamics."
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    "## Why use dynamical systems\n",
    "\n",
    "1. Because they (can) make a good model, and models are useful for\n",
    "  - predicting what happens next\n",
    "  - testing counterfactuals\n",
    "  - testing experimental hypotheses\n",
    "  - more specific to dynamical systems: they provide a *qualitative* understanding of a system's behavior\n",
    "2. Because they provide a **proof of concept** of the theoretical perspective described previously.\n",
    "\n",
    "Dynamical systems are general; just using a dynamical model doesn't bring in any assumptions (of course, you can use whatever assumptions you want in actually describing $\\mathbf{f}$).\n",
    "\n",
    "The biggest obstacle to a Gibsonian/interactivist perspective (particularly for someone coming from a cognitive background) on perception-action is plausibility.\n",
    "If we can show a dynamical system that contains only perceptual variables directly available to the agent, and the system behaves somewhat realistically, it shows that *control without a controller* is possible."
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    "## 1D example\n",
    "\n",
    "$$\\dot{x} = -kx$$"
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    "import numpy as np\n",
    "from ipywidgets import interact, FloatSlider\n",
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    "def f(x, k=1):\n",
    "    return -k * x"
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    "def init_plot(show_xaxis=True, show_yaxis=True):\n",
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    "\n",
    "\n",
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    "### Phase-space representation\n",
    "\n",
    "state $x$ along x axis, change in state $\\dot{x}$ along y axis.\n",
    "\n",
    "$k=1$\n",
    "\n",
    "Circles are *fixed points*. At a fixed point, state does not change.\n",
    "\n",
    "Filled circle = attractor. An attractor is a fixed point that attracts nearby states."
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       "            var render_items = [{\"docid\":\"27427091-a09a-4d02-8ff2-bb0fdf9e5919\",\"elementid\":\"b5cea01b-92aa-449e-9133-0239bdc99b40\",\"modelid\":\"92e904a7-f03d-43f8-b916-8d9c296d024a\",\"notebook_comms_target\":\"f8b82a64-e755-4f91-a140-e51220126e7c\"}];\n",
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       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = np.linspace(-2, 2, 50)\n",
    "\n",
    "attractor_fig = init_plot()\n",
    "attractor_fig.xaxis.axis_label = 'x'\n",
    "attractor_fig.yaxis.axis_label = 'x-dot'\n",
    "\n",
    "attractor_fig.line(xs, f(xs), line_width=2)\n",
    "attractor_fig.scatter(0, 0, size=20)\n",
    "add_arrow(attractor_fig, [-1.1, -0.9], 0)\n",
    "add_arrow(attractor_fig, [1.1, 0.9], 0)\n",
    "\n",
    "\n",
    "show(attractor_fig);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$k=1$\n",
    "\n",
    "Open circle = repeller = a fixed point that repels nearby states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "            var render_items = [{\"docid\":\"479b7ee7-3901-4001-8a92-69f4c4bae18b\",\"elementid\":\"97434987-b175-44f7-ab85-8af137789327\",\"modelid\":\"e4b3e02f-6f71-470e-90b2-48f70d24caab\",\"notebook_comms_target\":\"b47ade72-aeb9-4091-898e-93f85192ec8d\"}];\n",
       "            \n",
       "            Bokeh.embed.embed_items(docs_json, render_items);\n",
       "        });\n",
       "      },\n",
       "      function(Bokeh) {\n",
       "      }\n",
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       "    function run_inline_js() {\n",
       "      for (var i = 0; i < inline_js.length; i++) {\n",
       "        inline_js[i](window.Bokeh);\n",
       "      }\n",
       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "        run_inline_js();\n",
       "      });\n",
       "    }\n",
       "  }(this));\n",
       "</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "repeller_fig = init_plot()\n",
    "repeller_fig.xaxis.axis_label = 'x'\n",
    "repeller_fig.yaxis.axis_label = 'x-dot'\n",
    "\n",
    "repeller_fig.line(xs, f(xs, 1), line_width=2)\n",
    "repeller_fig.scatter(0, 0, size=20, fill_color='white', line_width=2)\n",
    "add_arrow(repeller_fig, [-0.9, -1.1], 0)\n",
    "add_arrow(repeller_fig, [0.9, 1.1], 0)\n",
    "\n",
    "\n",
    "show(repeller_fig);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Potential ($V$) representation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def v(x, k=1):\n",
    "    return k * x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "        Bokeh.$(function() {\n",
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       "            var render_items = [{\"docid\":\"3a7d7de6-ed43-437c-923b-53302545fcf9\",\"elementid\":\"0a5866eb-6a57-4f94-8cb2-8ff182a02903\",\"modelid\":\"0c749fe7-42a2-4bd8-adad-2963836942f7\",\"notebook_comms_target\":\"7f146c07-2092-416e-8963-fa63bc47ddc2\"}];\n",
       "            \n",
       "            Bokeh.embed.embed_items(docs_json, render_items);\n",
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       "      function(Bokeh) {\n",
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       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "        run_inline_js();\n",
       "      });\n",
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     "metadata": {},
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   ],
   "source": [
    "att_v_fig = init_plot(show_yaxis=False)\n",
    "attractor_fig.xaxis.visible = False\n",
    "att_v_fig.x_range = attractor_fig.x_range\n",
    "att_v_fig.xaxis.axis_label = 'x'\n",
    "att_v_fig.yaxis.axis_label = 'V'\n",
    "\n",
    "att_v_fig.line(xs, v(xs), line_width=2)\n",
    "att_v_fig.scatter(0, v(0), size=20)\n",
    "\n",
    "show(column([attractor_fig, att_v_fig]));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "        Bokeh.$(function() {\n",
       "            var docs_json = 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   "source": [
    "rep_v_fig = init_plot(show_yaxis=False)\n",
    "repeller_fig.xaxis.visible = False\n",
    "rep_v_fig.x_range = repeller_fig.x_range\n",
    "rep_v_fig.xaxis.axis_label = 'x'\n",
    "rep_v_fig.yaxis.axis_label = 'V'\n",
    "\n",
    "rep_v_fig.line(xs, v(xs, k=-1), line_width=2)\n",
    "rep_v_fig.scatter(0, v(0), size=20, line_width=2, fill_color='white')\n",
    "\n",
    "show(column([repeller_fig, rep_v_fig]));"
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   "source": [
    "### A nonlinear system\n",
    "\n",
    "$$ \\dot{x} = -rx - x^3 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def ddx(r, x):\n",
    "    return -r - 3 * x**2\n",
    "\n",
    "def dx(r, x):\n",
    "    return -r * x - x**3\n",
    "\n",
    "def v(r, x):\n",
    "    x = np.asarray(x)\n",
    "    return (1/2) * r * x**2 + (1/4) * x**4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
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       "            var render_items = [{\"docid\":\"35d22711-c926-4309-b8ac-8d42b0632c62\",\"elementid\":\"fd676109-7e18-4ffa-b3ae-9bcd50dd7346\",\"modelid\":\"9a5fdbd8-01d1-44c7-8987-685ea39c3890\",\"notebook_comms_target\":\"f65f059f-bec0-48a1-b3df-70f4e5f54266\"}];\n",
       "            \n",
       "            Bokeh.embed.embed_items(docs_json, render_items);\n",
       "        });\n",
       "      },\n",
       "      function(Bokeh) {\n",
       "      }\n",
       "    ];\n",
       "  \n",
       "    function run_inline_js() {\n",
       "      for (var i = 0; i < inline_js.length; i++) {\n",
       "        inline_js[i](window.Bokeh);\n",
       "      }\n",
       "    }\n",
       "  \n",
       "    if (window._bokeh_is_loading === 0) {\n",
       "      console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "      run_inline_js();\n",
       "    } else {\n",
       "      load_libs(js_urls, function() {\n",
       "        console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "        run_inline_js();\n",
       "      });\n",
       "    }\n",
       "  }(this));\n",
       "</script>"
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   "source": [
    "def roots(r, f):\n",
    "    attractors = []\n",
    "    repellers = []\n",
    "    halfs = []\n",
    "    all_roots = set()\n",
    "    for root in np.roots([-1, 0, -r, 0]):\n",
    "        if np.isreal(root):\n",
    "            root = float(np.real(root))\n",
    "            if root in all_roots:\n",
    "                continue\n",
    "            \n",
    "            all_roots.add(root)\n",
    "            sign = np.sign(f(r, root))\n",
    "            if sign == -1:\n",
    "                attractors.append(root)\n",
    "            elif sign == 1:\n",
    "                repellers.append(root)\n",
    "            elif sign == 0:\n",
    "                halfs.append(root)\n",
    "    return attractors, repellers, halfs\n",
    "\n",
    "\n",
    "r = 1.\n",
    "range_ = 4\n",
    "xs = np.linspace(-range_, range_, 1000)\n",
    "\n",
    "fig = init_plot()\n",
    "fig.xaxis.visible = False\n",
    "fig.yaxis.axis_label = 'x-dot'\n",
    "dx_glyph = fig.line(xs, dx(r, xs), line_width=2)\n",
    "attractors, repellers, halfs = roots(r, ddx)\n",
    "att_glyph = fig.scatter(attractors, np.zeros_like(attractors), size=20)\n",
    "rep_glyph = fig.scatter(repellers, np.zeros_like(repellers), size=20, line_width=2, fill_color='white')\n",
    "half_glyphs = fig.scatter(halfs, np.zeros_like(halfs), size=20, line_width=2, fill_alpha=0.5)\n",
    "\n",
    "v_fig = init_plot(show_yaxis=False)\n",
    "v_fig.xaxis.axis_label = 'x'\n",
    "v_fig.yaxis.axis_label = 'V'\n",
    "v_fig.x_range = fig.x_range\n",
    "v_glyph = v_fig.line(xs, v(r, xs), line_width=2)\n",
    "#v_att_glyphs = v_fig.scatter(attractors, v(r, attractors), size=20)\n",
    "#v_rep_glyphs = v_fig.scatter(repellers, v(r, repellers), size=20, line_width=2, fill_color='white')\n",
    "#v_half_glyphs = v_fig.scatter(halfs, v(r, halfs), size=20, line_width=2, fill_alpha=0.5)\n",
    "\n",
    "show(column([fig, v_fig]));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
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   "source": [
    "@interact(r=FloatSlider(min=-10., max=10., value=1.))\n",
    "def update(r):\n",
    "    dx_glyph.data_source.data['y'] = dx(r, xs)\n",
    "    v_glyph.data_source.data['y'] = v(r, xs)\n",
    "    attractors, repellers, halfs = roots(r, ddx)\n",
    "    att_glyph.data_source.data = {'x': attractors, 'y': np.zeros_like(attractors)}\n",
    "    rep_glyph.data_source.data = {'x': repellers, 'y': np.zeros_like(repellers)}\n",
    "    half_glyphs.data_source.data = {'x': halfs, 'y': np.zeros_like(halfs)}\n",
    "    #v_att_glyphs.data_source.data = {'x': attractors, 'y': v(r, attractors)}\n",
    "    #v_rep_glyphs.data_source.data = {'x': repellers, 'y': v(r, repellers)}\n",
    "    #v_half_glyphs.data_source.data = {'x': halfs, 'y': v(r, halfs)}\n",
    "    push_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A 2D system: Damped spring"
   ]
  },
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   "metadata": {},
   "source": [
    "Dynamics are a useful modeling tool because they allow us to develop *spatial intuitions* about our systems in order to get a qualitative view of their behavior."
   ]
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   "execution_count": 90,
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       "<img src=\"https://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif\" alt=\"mass-spring\" style=\"width:640px;height:400px;\">"
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    "%%HTML\n",
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   "source": [
    "$$ m\\ddot{x} + b\\dot{x} + kx = 0 $$\n",
    "\n",
    "or as a system of first-order differential equations\n",
    "\n",
    "\\begin{aligned}\n",
    "\\dot{x} &= v \\\\\n",
    "\\dot{v} &= -bv -x\n",
    "\\end{aligned}\n",
    "\n",
    "with mass $m$, damping strength $b$, position $x$, velocity $v$."
   ]
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  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": [
    "def dv(b, x, v):\n",
    "    return -b * v - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
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    "def streamlines(x, y, u, v, density=1):\n",
    "    ''' Return streamlines of a vector flow.\n",
    "\n",
    "    * x and y are 1d arrays defining an *evenly spaced* grid.\n",
    "    * u and v are 2d arrays (shape [y,x]) giving velocities.\n",
    "    * density controls the closeness of the streamlines. For different\n",
    "      densities in each direction, use a tuple or list [densityx, densityy].\n",
    "\n",
    "    '''\n",
    "\n",
    "    ## Set up some constants - size of the grid used.\n",
    "    NGX = len(x)\n",
    "    NGY = len(y)\n",
    "\n",
    "    ## Constants used to convert between grid index coords and user coords.\n",
    "    DX = x[1]-x[0]\n",
    "    DY = y[1]-y[0]\n",
    "    XOFF = x[0]\n",
    "    YOFF = y[0]\n",
    "\n",
    "    ## Now rescale velocity onto axes-coordinates\n",
    "    u = u / (x[-1]-x[0])\n",
    "    v = v / (y[-1]-y[0])\n",
    "    speed = np.sqrt(u*u+v*v)\n",
    "    ## s (path length) will now be in axes-coordinates, but we must\n",
    "    ## rescale u for integrations.\n",
    "    u *= NGX\n",
    "    v *= NGY\n",
    "    ## Now u and v in grid-coordinates.\n",
    "\n",
    "    NBX = int(30*density)\n",
    "    NBY = int(30*density)\n",
    "    blank = np.zeros((NBY,NBX))\n",
    "\n",
    "    bx_spacing = NGX/float(NBX-1)\n",
    "    by_spacing = NGY/float(NBY-1)\n",
    "\n",
    "    def blank_pos(xi, yi):\n",
    "        return int((xi / bx_spacing) + 0.5), \\\n",
    "               int((yi / by_spacing) + 0.5)\n",
    "\n",
    "    def value_at(a, xi, yi):\n",
    "        if type(xi) == np.ndarray:\n",
    "            x = xi.astype(np.int)\n",
    "            y = yi.astype(np.int)\n",
    "        else:\n",
    "            x = np.int(xi)\n",
    "            y = np.int(yi)\n",
    "        a00 = a[y,x]\n",
    "        a01 = a[y,x+1]\n",
    "        a10 = a[y+1,x]\n",
    "        a11 = a[y+1,x+1]\n",
    "        xt = xi - x\n",
    "        yt = yi - y\n",
    "        a0 = a00*(1-xt) + a01*xt\n",
    "        a1 = a10*(1-xt) + a11*xt\n",
    "        return a0*(1-yt) + a1*yt\n",
    "\n",
    "    def rk4_integrate(x0, y0):\n",
    "        ## This function does RK4 forward and back trajectories from\n",
    "        ## the initial conditions, with the odd 'blank array'\n",
    "        ## termination conditions. TODO tidy the integration loops.\n",
    "\n",
    "        def f(xi, yi):\n",
    "            dt_ds = 1./value_at(speed, xi, yi)\n",
    "            ui = value_at(u, xi, yi)\n",
    "            vi = value_at(v, xi, yi)\n",
    "            return ui*dt_ds, vi*dt_ds\n",
    "\n",
    "        def g(xi, yi):\n",
    "            dt_ds = 1./value_at(speed, xi, yi)\n",
    "            ui = value_at(u, xi, yi)\n",
    "            vi = value_at(v, xi, yi)\n",
    "            return -ui*dt_ds, -vi*dt_ds\n",
    "\n",
    "        check = lambda xi, yi: xi>=0 and xi<NGX-1 and yi>=0 and yi<NGY-1\n",
    "\n",
    "        bx_changes = []\n",
    "        by_changes = []\n",
    "\n",
    "        ## Integrator function\n",
    "        def rk4(x0, y0, f):\n",
    "            ds = 0.01 #min(1./NGX, 1./NGY, 0.01)\n",
    "            stotal = 0\n",
    "            xi = x0\n",
    "            yi = y0\n",
    "            xb, yb = blank_pos(xi, yi)\n",
    "            xf_traj = []\n",
    "            yf_traj = []\n",
    "            while check(xi, yi):\n",
    "                # Time step. First save the point.\n",
    "                xf_traj.append(xi)\n",
    "                yf_traj.append(yi)\n",
    "                # Next, advance one using RK4\n",
    "                try:\n",
    "                    k1x, k1y = f(xi, yi)\n",
    "                    k2x, k2y = f(xi + .5*ds*k1x, yi + .5*ds*k1y)\n",
    "                    k3x, k3y = f(xi + .5*ds*k2x, yi + .5*ds*k2y)\n",
    "                    k4x, k4y = f(xi + ds*k3x, yi + ds*k3y)\n",
    "                except IndexError:\n",
    "                    # Out of the domain on one of the intermediate steps\n",
    "                    break\n",
    "                xi += ds*(k1x+2*k2x+2*k3x+k4x) / 6.\n",
    "                yi += ds*(k1y+2*k2y+2*k3y+k4y) / 6.\n",
    "                # Final position might be out of the domain\n",
    "                if not check(xi, yi): break\n",
    "                stotal += ds\n",
    "                # Next, if s gets to thres, check blank.\n",
    "                new_xb, new_yb = blank_pos(xi, yi)\n",
    "                if new_xb != xb or new_yb != yb:\n",
    "                    # New square, so check and colour. Quit if required.\n",
    "                    if blank[new_yb,new_xb] == 0:\n",
    "                        blank[new_yb,new_xb] = 1\n",
    "                        bx_changes.append(new_xb)\n",
    "                        by_changes.append(new_yb)\n",
    "                        xb = new_xb\n",
    "                        yb = new_yb\n",
    "                    else:\n",
    "                        break\n",
    "                if stotal > 2:\n",
    "                    break\n",
    "            return stotal, xf_traj, yf_traj\n",
    "\n",
    "        integrator = rk4\n",
    "\n",
    "        sf, xf_traj, yf_traj = integrator(x0, y0, f)\n",
    "        sb, xb_traj, yb_traj = integrator(x0, y0, g)\n",
    "        stotal = sf + sb\n",
    "        x_traj = xb_traj[::-1] + xf_traj[1:]\n",
    "        y_traj = yb_traj[::-1] + yf_traj[1:]\n",
    "\n",
    "        ## Tests to check length of traj. Remember, s in units of axes.\n",
    "        if len(x_traj) < 1: return None\n",
    "        if stotal > .2:\n",
    "            initxb, inityb = blank_pos(x0, y0)\n",
    "            blank[inityb, initxb] = 1\n",
    "            return x_traj, y_traj\n",
    "        else:\n",
    "            for xb, yb in zip(bx_changes, by_changes):\n",
    "                blank[yb, xb] = 0\n",
    "            return None\n",
    "\n",
    "    ## A quick function for integrating trajectories if blank==0.\n",
    "    trajectories = []\n",
    "    def traj(xb, yb):\n",
    "        if xb < 0 or xb >= NBX or yb < 0 or yb >= NBY:\n",
    "            return\n",
    "        if blank[yb, xb] == 0:\n",
    "            t = rk4_integrate(xb*bx_spacing, yb*by_spacing)\n",
    "            if t is not None:\n",
    "                trajectories.append(t)\n",
    "\n",
    "    ## Now we build up the trajectory set. I've found it best to look\n",
    "    ## for blank==0 along the edges first, and work inwards.\n",
    "    for indent in range((max(NBX,NBY))//2):\n",
    "        for xi in range(max(NBX,NBY)-2*indent):\n",
    "            traj(xi+indent, indent)\n",
    "            traj(xi+indent, NBY-1-indent)\n",
    "            traj(indent, xi+indent)\n",
    "            traj(NBX-1-indent, xi+indent)\n",
    "\n",
    "    xs = [np.array(t[0])*DX+XOFF for t in trajectories]\n",
    "    ys = [np.array(t[1])*DY+YOFF for t in trajectories]\n",
    "\n",
    "    return xs, ys\n",
    "\n",
    "xx = np.linspace(-3, 3, 100)\n",
    "yy = np.linspace(-3, 3, 100)\n",
    "\n",
    "Y, X = np.meshgrid(xx, yy)\n",
    "U = Y\n",
    "V = dv(1, X, Y)\n",
    "speed = np.sqrt(U*U + V*V)\n",
    "theta = np.arctan(V/U)\n",
    "\n",
    "x0 = X[::2, ::2].flatten()\n",
    "y0 = Y[::2, ::2].flatten()\n",
    "length = speed[::2, ::2].flatten()/40\n",
    "angle = theta[::2, ::2].flatten()\n",
    "x1 = x0 + length * np.cos(angle)\n",
    "y1 = y0 + length * np.sin(angle)\n",
    "\n",
    "xs, ys = streamlines(xx, yy, U.T, V.T, density=1)\n",
    "\n",
    "cm = np.array([\"#C7E9B4\", \"#7FCDBB\", \"#41B6C4\", \"#1D91C0\", \"#225EA8\", \"#0C2C84\"])\n",
    "ix = ((length-length.min())/(length.max()-length.min())*5).astype('int')\n",
    "colors = cm[ix]\n",
    "\n",
    "p1 = figure(x_range=(-3,3 ), y_range=(-3, 3))\n",
    "quiver = p1.segment(x0, y0, x1, y1, color=colors, line_width=2)\n",
    "\n",
    "multi_line = p1.multi_line(xs, ys, color=\"#ee6666\", line_width=2, line_alpha=0.8)\n",
    "p1.grid.visible = False\n",
    "p1.xaxis.axis_label = 'x'\n",
    "p1.yaxis.axis_label = 'v'\n",
    "p1.add_layout(Span(location=0, dimension='height', line_width=1, line_color='black'))\n",
    "p1.add_layout(Span(location=0, dimension='width', line_width=1, line_color='black'))\n",
    "p1.scatter(0, 0, size=20, color=\"#ee6666\")\n",
    "\n",
    "show(p1);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@interact(b=FloatSlider(min=0., max=2., value=0.))\n",
    "def update(b):\n",
    "    V = dv(b, X, Y)\n",
    "    speed = np.sqrt(U*U + V*V)\n",
    "    theta = np.arctan(V/U)\n",
    "\n",
    "    length = speed[::2, ::2].flatten()/40\n",
    "    angle = theta[::2, ::2].flatten()\n",
    "    x1 = x0 + length * np.cos(angle)\n",
    "    y1 = y0 + length * np.sin(angle)\n",
    "    xs, ys = streamlines(xx, yy, U.T, V.T, density=1)\n",
    "\n",
    "    cm = np.array([\"#C7E9B4\", \"#7FCDBB\", \"#41B6C4\", \"#1D91C0\", \"#225EA8\", \"#0C2C84\"])\n",
    "    ix = ((length-length.min())/(length.max()-length.min())*5).astype('int')\n",
    "    colors = cm[ix]\n",
    "    \n",
    "    quiver.data_source.data = dict(x0=x0, y0=y0, x1=x1, y1=y1)\n",
    "    multi_line.data_source.data = dict(xs=xs, ys=ys)\n",
    "    push_notebook()\n",
    "\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### >2 dimensions\n",
    "\n",
    "Chaos = sensitivity to initial conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src=\"https://upload.wikimedia.org/wikipedia/commons/4/45/Double-compound-pendulum.gif\" alt=\"double-pendulum\" style=\"width:400;height:400px;\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "%%HTML\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/4/45/Double-compound-pendulum.gif\" alt=\"double-pendulum\" style=\"width:400;height:400px;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sociodynamics\n",
    "\n",
    "![](http://i.imgur.com/GPGTD1T.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Back to behavioral dynamics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](http://i.imgur.com/t29qPXk.png)\n",
    "\n",
    "![](http://i.imgur.com/ExVKw5e.png)\n",
    "\n",
    "![](http://i.imgur.com/mJVcuig.png)\n",
    "\n",
    "![](http://i.imgur.com/XuxfN7P.png)\n",
    "\n",
    "![](http://i.imgur.com/pCGecWX.png)"
   ]
  }
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