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   "execution_count": 1,
   "id": "eaa65a42-a9a7-4420-acb7-4fee6610fa5b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sp \n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a78f18ca-7b95-4f82-b501-ef9f402ecbc7",
   "metadata": {},
   "source": [
    "Dane Sabo\n",
    "October 2nd, 2024\n",
    "\n",
    "# Problem 3.7.2 (Written)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0af0982d-25ae-4a1d-82d3-262935ab1d81",
   "metadata": {},
   "outputs": [],
   "source": [
    "def linear_phase_portrait_info(A):\n",
    "    print('\\n Input Matrix:')\n",
    "    print(A)\n",
    "    \n",
    "    print('\\n Eigenvalues and Eigenvectors:')\n",
    "    print(np.linalg.eig(A).eigenvalues)\n",
    "    print(np.linalg.eig(A).eigenvectors)\n",
    "    \n",
    "    print('\\n Trace:')\n",
    "    print(np.linalg.trace(A))\n",
    "\n",
    "    print('\\n Determinant:')\n",
    "    print(np.linalg.det(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bafdbc61-fba6-490b-8793-6b1820d6fbd4",
   "metadata": {},
   "source": [
    "Sketch phase portraits for the following linear systems:\n",
    "## $\\dot x = 0$, $\\dot y = x + 2y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "37b37e65-7ade-473e-82ff-6391c6c6fe1d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Input Matrix:\n",
      "[[0 0]\n",
      " [1 2]]\n",
      "\n",
      " Eigenvalues and Eigenvectors:\n",
      "[2. 0.]\n",
      "[[ 0.          0.89442719]\n",
      " [ 1.         -0.4472136 ]]\n",
      "\n",
      " Trace:\n",
      "2\n",
      "\n",
      " Determinant:\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[0,0],[1,2]])\n",
    "linear_phase_portrait_info(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "966ca5a8-bcf4-4860-b923-f5cca9f6e1de",
   "metadata": {},
   "source": [
    "## $\\dot x = x+2y$, $\\dot y = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "6e05a8d7-7bf6-4520-a3b6-0848d067b189",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Input Matrix:\n",
      "[[1 2]\n",
      " [0 0]]\n",
      "\n",
      " Eigenvalues and Eigenvectors:\n",
      "[1. 0.]\n",
      "[[ 1.         -0.89442719]\n",
      " [ 0.          0.4472136 ]]\n",
      "\n",
      " Trace:\n",
      "1\n",
      "\n",
      " Determinant:\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1,2],[0,0]])\n",
    "linear_phase_portrait_info(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0179406c-daa5-47bf-a10f-126f1ffe92c4",
   "metadata": {},
   "source": [
    "## $\\dot x = 3x+4y$, $\\dot y = 4x - 3y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0c198580-fd00-43d8-aeb7-947c88f449c0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Input Matrix:\n",
      "[[ 3  4]\n",
      " [ 4 -3]]\n",
      "\n",
      " Eigenvalues and Eigenvectors:\n",
      "[ 5. -5.]\n",
      "[[ 0.89442719 -0.4472136 ]\n",
      " [ 0.4472136   0.89442719]]\n",
      "\n",
      " Trace:\n",
      "0\n",
      "\n",
      " Determinant:\n",
      "-25.000000000000007\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[3,4],[4,-3]])\n",
    "linear_phase_portrait_info(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a41e8548-555b-4701-bad7-9049cb395dec",
   "metadata": {},
   "source": [
    "## $\\dot x = 3x+y$, $\\dot y = -x + 3y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "40cd3d5a-a712-4fab-94bb-8148d2fd61ad",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Input Matrix:\n",
      "[[ 3  1]\n",
      " [-1  3]]\n",
      "\n",
      " Eigenvalues and Eigenvectors:\n",
      "[3.+1.j 3.-1.j]\n",
      "[[0.70710678+0.j         0.70710678-0.j        ]\n",
      " [0.        +0.70710678j 0.        -0.70710678j]]\n",
      "\n",
      " Trace:\n",
      "6\n",
      "\n",
      " Determinant:\n",
      "10.000000000000002\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[3,1],[-1,3]])\n",
    "linear_phase_portrait_info(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cdd8c34-e628-4bce-b37e-74f0da7a7bfd",
   "metadata": {},
   "source": [
    "# Problem 3.7.4 (Python)\n",
    "Plot the phase portraits for the following systems:\n",
    "## $\\dot x = y$, $\\dot y = x-y+x^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21643903-10a1-47ca-b449-4dcb6a4819e7",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "c6c01322-669f-4bd4-8727-addd978c85e2",
   "metadata": {},
   "source": [
    "## $\\dot x = -2x-y+2$, $\\dot y = xy$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cd100989-95c4-4d6e-87a2-23468c3e7d30",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "91f924f6-1567-47c9-893a-93cbf5753b98",
   "metadata": {},
   "source": [
    "## $\\dot x = x^2- y^2$, $\\dot y = xy-1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d9c0e844-0458-4025-87a6-2b7f0332e8f5",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "312d3ea7-a2c1-4db3-9aef-d4882adba1f3",
   "metadata": {},
   "source": [
    "## $\\dot x = 2-x-y^2$, $\\dot y = -y(x^2 + y^2 -3x +1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cfb3f6a8-fcf2-4669-a566-21ceea20b5ca",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "12ad95a7-06dd-45b9-8d48-4a7e0451422e",
   "metadata": {},
   "source": [
    "# Problem 3.7.5 (Written)\n",
    "Construct a nonlinear system that has four critical points: two saddle points, one stable focus, and one unstable focus."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf8225d4-cedb-4daa-a7b0-ef9b032c4c92",
   "metadata": {},
   "source": [
    "# Problem 3.7.6 (Written)\n",
    "A nonlinear capacitor-resistor electrical circuit can be modeled using the differential equations\n",
    "$$ \\dot x = y $$\n",
    "$$\\dot y = -x + x^3 - (a_0 + x)y$$\n",
    "where $a_0$ is a nonzero constant and $x(t)$ represents the current in the circuit at time t. Sketch phase portraits when $a_0 > 0$ and $a_0<0$ and give a physical interpretation of the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "737e53cb-5f98-42b4-9ca5-696cc03cc724",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "66d21987-bc58-4f31-af7b-94bcaf967921",
   "metadata": {},
   "source": [
    "# Problem 4.5.6 (Python)\n",
    "A predator-prey system may be modeled using the differential equations\n",
    "$$ \\dot x = x (1-y-\\epsilon x)$$\n",
    "$$ \\dot y = y (-1 + x - \\epsilon y)$$\n",
    "where $x(t)$ is the population of prey and $y(t)$ is the predator population size at time t, respectively. Classify the critical points for $\\epsilon>0$ and plot phase portraits for the different types of qualitative behavior. Interpret the results in physical terms"
   ]
  }
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