I am trying to understand a simple property of odes defined on a real line (or a subset of the real line).
Consider an ordinary autonomous differential equation where the dependent variable $x$ is defined on the real line, i.e. $$\dot{x} = f(x),$$ where $x \in \mathbb{R}$, and the dot is differentiation with respect to time $t$. It is known that the corresponding phase portrait has solutions that are equilibria, and all other solutions either move towards $\pm \infty$, or towards / away from the equilibria. Geometrically, if we draw the phase portrait, we only can have equilibria and all other trajectories move right or left. No periodic solutions exist.
So what about nonautonomous ode like $$\dot{x}=\cos t.$$ The general solution is $x(t)=\sin t - \sin t_{0} +x_{0}$. It is clear that all solutions are periodic. Does that contradict the above theory? How would the corresponding phase portrait look like?
You can transform every nonautonomous ODE to a autonomous one by introducing a new variable $z$ with $\dot z = 1$ and replacing $t$ by $z$. But this will always add a dimension to the system. Hence it does not contradict the theory, because it only considers the case $$\dot x = f(x)$$ which is an autonomous ODE on $\mathbb R$ and not $\mathbb R^k$ for $k>1$ and not nonautonomous ODEs.