typos fixes

Author: f14-bertolotti

posts/15-12-24-chaos-I/index.html

@@ -97,7 +97,7 @@ <h3> Preliminaries </h3>
             </p>
 
             <p>
-                More intuitively, you can think of a manifold as an object embedded in a Euclidean space. For example, imagine a circle of radius 1 in the Euclidean plane. You are looking at a 1-dimensional manifold. Why 1-dimensional? Because you can describe every point on the circle with a single parameter: the angle. Or think of a hollow sphere. You are looking at a 2-dimensional manifold. Why 2-dimensional? Because you can describe every point on the sphere with two parameters: latitude and longitude. This means you can map the circle to a line and the sphere to a plane.
+                More intuitively, you can think of a manifold as an object embedded in a Euclidean space. For example, imagine a circle of radius 1 in the Euclidean plane. You are looking at a 1-dimensional manifold. Why 1-dimensional? Because you can describe every point on the circle with a single parameter: the angle. Or think of a hollow sphere. You are looking at a 2-dimensional manifold. Why 2-dimensional? Because you can describe every point on the sphere with two parameters: latitude and longitude. 
             </p>
 
             <p>
@@ -339,7 +339,7 @@ <h3>Autonomous and Non-Autonomous Systems</h3>
             </p>
 
             <p>
-                The solution to the initial value problem is given by a function \(f(x_0,t_0,t)\), which maps the initial state \(x_0\) and the initial time \(t_0\) to the state at time \(t\). Then, we have that \(f(x_0,t_0,0) = x_0\). Additionally, we can verify that:
+                The solution to the initial value problem is given by a function \(f(x_0,t_0,t)\), which maps the initial state \(x_0\) and the initial time \(t_0\) to the state at time \(t\). Then, we have that \(f(x_0,t_0,0) = x_0,t_0\). Additionally, we can verify that:
                 \[
                 f(f(x_0,t_0,t_1),t_2) = f(f(x_0,t_0+t_1,0),t_2) = f(x_0,t_0+t_1,t_2) = f(x_0,t_0,t_1+t_2)
                 \]