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   "source": [
    "### **Course**: BIO-341 [_Dynamical systems in biology_](https://moodle.epfl.ch/course/info.php?id=14291)\n",
    "\n",
    "**Professor**: _Julian Shillcock_ & _Felix Naef_\n",
    "\n",
    "SSV, BA5, 2025"
   ]
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib_inline.backend_inline\n",
    "from ipywidgets import interact\n",
    "from scipy.integrate import odeint\n",
    "from IPython.display import set_matplotlib_formats\n",
    "matplotlib_inline.backend_inline.set_matplotlib_formats('png', 'pdf')"
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   "source": [
    "# One-dimensional bifurcations (pencil and paper)"
   ]
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   "source": [
    "## Question 1\n",
    "For each of the 1D vector fields A – G:\n",
    "\n",
    "A) $\\frac{dx}{dt} = r - x^2$\n",
    "\n",
    "B) $\\frac{dx}{dt} = rx - x^2$ \n",
    "\n",
    "C) $\\frac{dx}{dt} = rx - 4x^3$ \n",
    "\n",
    "D) $\\frac{dx}{dt} = rx + 4x^3$ \n",
    "\n",
    "E) $\\frac{dx}{dt} = rx - sinh(x)$  \n",
    "\n",
    "F) $\\frac{dx}{dt} = x + \\frac{rx}{1 + x^2}$  \n",
    "\n",
    "1.1 Sketch the vector field for the three cases r < 0, r = 0, r > (labelled by their value of r), mark the fixed point(s) and their stability, and indicate the direction of flow of the vector field on the diagram.\n",
    "\n",
    "1.2 Sketch a few typical trajectories for the three cases, r < 0, r = 0, r > 0.\n",
    "\n",
    "1.3 Draw the bifurcation diagram and label the branches as stable or unstable (use a solid line for stable branches and a dashed line for unstable branches).\n",
    "\n",
    "1.4 Identify the type of bifurcation in the bifurcation diagram.\n"
   ]
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   "source": [
    "## Question 2\n",
    "Construct a vector field that depends on a real parameter r, dx/dt = f(r, x), and that has no fixed points for r < 0, and four fixed points for r > 0. Sketch the vector field in the two cases, indicate the direction of flow, and label any fixed points with their stability.  \n",
    "\n",
    "\n"
   ]
  },
  {
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   "id": "b5e606b8-faae-4891-bd20-f87163397437",
   "metadata": {},
   "source": [
    "## Question 3\n",
    "Draw the bifurcation diagram for the system in Question 2.\n"
   ]
  }
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