$x ^ { \prime } = \cos \left( x ^ { 2 } \right)$
Given the above equation, I need to show that it determines a dynamical system.
So, since this cannot be directly solved, I tried using Lipschitz to prove that each solution is contuinable on $R$.
Thus, I defined a function $f(x)$ as follows.
$f ( x ) = \cos \left( x ^ { 2 } \right)$
$f ^ { \prime } ( x ) = - 2 x \cdot \sin \left( x ^ { 2 } \right)$
I have taken two arbitrary points $x_1, x_2$ and applied mean value theorem on $(x_2,x_1)$.
$\frac { f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) } { x _ { 2 } - x _ { 1 } } = f ^ { \prime } ( c )$
$= > \quad \left| f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) \right| = \left| x _ { 2 } - x _ { 1 } \right| f ^ { \prime } ( x )$
$= > \left| f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) \right| = | - 2 c | \cdot \left| \sin \left( c ^ { 2 } \right) \right|$
However, the right hand side has no supremum on $R$. I got stuck at that point.