chap 6 updated.

docs/examples/chap_06.ipynb

@@ -85,16 +85,17 @@
     "\n",
     "#### Step 2: Find the Nullclines\n",
     "\n",
-    "- **Nullcline for $ \\frac{dx}{dt} = 0 $**: Set $ f(x, y) = 0 $.\n",
-    "- **Nullcline for $ \\frac{dy}{dt} = 0 $**: Set $ g(x, y) = 0 $.\n",
+    "- Nullcline for  $\\frac{dx}{dt} = 0$ : Set  $f(x, y) = 0$ .\n",
+    "- Nullcline for  $\\frac{dy}{dt} = 0$ : Set  $g(x, y) = 0$ .\n",
+    "\n",
     "\n",
     "#### Step 3: Use SymPy to Solve for Nullclines\n",
     "\n",
-    "You can use `SymPy` to symbolically solve for $ y $ in terms of $x$ (or vice versa) where these equations are zero.\n",
+    "You can use `SymPy` to symbolically solve for $y$ in terms of $x$ (or vice versa) where these equations are zero.\n",
     "\n",
     "#### Step 4: Plot the Nullclines Using Matplotlib\n",
     "\n",
-    "After finding the expressions for the nullclines, you can use `Matplotlib` to plot them over a grid of values for $ x $ and $ y $.\n",
+    "After finding the expressions for the nullclines, you can use `Matplotlib` to plot them over a grid of values for $x$ and $y$.\n",
     "\n",
     "#### Example\n",
     "\n",
@@ -109,11 +110,11 @@
     "\n",
     "#### Explanation\n",
     "\n",
-    "1. **Defining the system**: The functions $ f(x, y) $ and $ g(x, y) $ represent the right-hand side of the system of differential equations.\n",
+    "1. **Defining the system**: The functions $f(x, y)$ and $g(x, y)$ represent the right-hand side of the system of differential equations.\n",
     "\n",
-    "2. **Finding nullclines**: We solve $ f(x, y) = 0 $ and $ g(x, y) = 0 $ for $ y $.\n",
+    "2. **Finding nullclines**: We solve $f(x, y) = 0$ and $g(x, y) = 0$ for $y$.\n",
     "\n",
-    "3. **Plotting**: The nullclines are plotted using `Matplotlib`, with one set for $ \\frac{dx}{dt} = 0 $ and another for $ \\frac{dy}{dt} = 0 $.\n",
+    "3. **Plotting**: The nullclines are plotted using `Matplotlib`, with one set for $\\frac{dx}{dt} = 0$ and another for $\\frac{dy}{dt} = 0$.\n",
     "\n",
     "#### Tools Used\n",
     "\n",
@@ -226,7 +227,7 @@
    "metadata": {},
    "source": [
     "### adding an interactive trajectory solution\n",
-    "If you are getting messy output just comment the line @iteract()\n"
+    "If you are getting messy output just comment the line iteract()\n"
    ]
   },
   {
@@ -363,8 +364,8 @@
     "### Explanation:\n",
     "\n",
     "1. **Step 1 (Equilibrium Points):** \n",
-    "   - We define a system of equations $ \\frac{dX}{dt} = F(X) $. \n",
-    "   - The equilibrium points are found by solving $ F(X) = 0 $.\n",
+    "   - We define a system of equations $\\frac{dX}{dt} = F(X)$. \n",
+    "   - The equilibrium points are found by solving $F(X) = 0$.\n",
     "\n",
     "2. **Step 2 (Jacobian Matrix):**\n",
     "   - The Jacobian matrix is calculated symbolically using `sympy.jacobian`.\n",
@@ -803,7 +804,7 @@
     "\\end{align*}\n",
     "$$\n",
     "\n",
-    "where $\\tau = 20 $ ms, $\\tau_a = 4000$ ms.\n",
+    "where $\\tau = 20$ ms, $\\tau_a = 4000$ ms.\n",
     "\n",
     "for plotting the isoclines, we consider the first two equations and consider $A_1$ as a parameter."
    ]