Nov 11 -  The Logistics Map.ipynb

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    "#  The Logistics Map\n",
    "\n",
    "# Poincare sections show us how to exploit the periodicity of the driving term in order to find some \"order\" out of \"chaos\"\n",
    "\n",
    "# The \"logistics map\" is a way to quantify this sort of recurring behavior\n",
    "\n",
    "# My apologies, but we will just take a quick look at this. Lots more in Taylor, and elsewhere\n",
    "\n",
    "# Consider the \"map\" x(n+1) = r x(n)(1-x(n)) ..."
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    "# Taylor Figure 12.34\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Define the recurrence equation\n",
    "def recurrence_equation(x, r):\n",
    "    return r * x * (1 - x)\n",
    "\n",
    "# Set the parameters\n",
    "r = 0.8\n",
    "n_max = 12\n",
    "\n",
    "# Set the initial conditions\n",
    "x0p1 = [0.1]\n",
    "x0p5 = [0.5]\n",
    "\n",
    "# Generate the sequence for x0p1\n",
    "for n in range(n_max):\n",
    "    x0p1.append(recurrence_equation(x0p1[-1], r))\n",
    "\n",
    "# Generate the sequence for x0p5\n",
    "for n in range(n_max):\n",
    "    x0p5.append(recurrence_equation(x0p5[-1], r))\n",
    "\n",
    "# Plot the list plot\n",
    "plt.plot(range(n_max + 1), x0p1, marker='o', label='x0p1')\n",
    "plt.plot(range(n_max + 1), x0p5, marker='o', label='x0p5')\n",
    "plt.xlabel('n')\n",
    "plt.ylabel('x')\n",
    "plt.title('List Plot')\n",
    "plt.legend()\n",
    "plt.show()\n",
    "\n",
    "\n",
    "# Now, see what happens for r=1.5, 2.5, 3.5"
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    "# What is going on here?\n",
    "\n",
    "# As n goes to infinity we expect x to approach a \"fixed point\"\n",
    "\n",
    "\n",
    "# Define the function\n",
    "def logistic_function(x, r):\n",
    "    return r * x * (1 - x)\n",
    "\n",
    "# Set the parameter\n",
    "r = 0.8\n",
    "\n",
    "# Generate x values\n",
    "x = np.linspace(0, 1, 100)\n",
    "\n",
    "# Evaluate the function\n",
    "y1 = x\n",
    "y2 = logistic_function(x, r)\n",
    "\n",
    "# Plot the functions\n",
    "plt.plot(x, y1, label='x', linestyle='-', linewidth=2)\n",
    "plt.plot(x, y2, label='r x (1 - x)', linestyle='-', linewidth=2)\n",
    "\n",
    "# Set plot labels and title\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Plot')\n",
    "plt.legend()\n",
    "\n",
    "# Display the plot\n",
    "plt.show()\n"
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    "# Stability of a fixed point x*\n",
    "\n",
    "# For x(n+1)=f[x(n)], consider motion for small deviation from a fixed point\n",
    "\n",
    "#### x(n+1)=f(x*+\\[Epsilon])\\[TildeTilde]f(x*)+f'(x*)\\[Epsilon]=x*+f'(x*)\\[Epsilon]\n",
    "\n",
    "# So if |f'(x*)|<1, successive iterations converge and the fixed point is stable\n",
    "\n",
    "import sympy as sp\n",
    "\n",
    "# Define the function\n",
    "def f(x, r):\n",
    "    return r * x * (1 - x)\n",
    "\n",
    "# Set the parameter\n",
    "r = 0.8\n",
    "\n",
    "# Solve the equation f(x) = x\n",
    "x = sp.symbols('x')\n",
    "solution = sp.solve(f(x, r) - x, x)\n",
    "\n",
    "# Calculate the derivative f'(x) and substitute the solution\n",
    "derivative = sp.diff(f(x, r), x)\n",
    "derivative_solution = derivative.subs(x, solution[0])\n",
    "\n",
    "# Print the solutions and derivative value\n",
    "print(\"Solutions:\", solution)\n",
    "print(\"f'(x) =\", derivative_solution)\n",
    "\n",
    "# Find no stable fixed points if r>3"
   ]
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