Let $M$ be a smooth connected manifold. For every initial condition $(p,v) \in M \times T_{p}M$, suppose that a given system of (second order) ordinary differential equations has unique global solution, i.e. the solution is unique and exists for all time.
Does this imply that any two point of the manifold can be joined by a solution of the given ODE system?
EDIT: Since every solution starting at $p \in M$ is defined for all time, we can define an "exponential map" $\mathsf{exp}_{p}$, which takes as input a vector $v \in T_{p}M$ and returns the point $\gamma(1) \in M$. Here $\gamma$ is the (unit-speed) solution starting at $p$ with initial tangent vector $v$. By assumption, it is clear that $\mathsf{exp}_{p}$ is well-defined on the entire $T_{p}M$. My question is whether it is also surjective.
The answer is no: global existence of solutions does not ensure this kind of "transitivity".
As a counterexample in $\mathbb{R}^2$, consider a system akin to a particle with velocity dependent charge in a uniform magnetic field: $$\begin{align} \ddot{x}&=-\dot{y}\sqrt{\dot{x}^2+\dot{y}^2} \\ \ddot{y}&=\dot{x}\sqrt{\dot{x}^2+\dot{y}^2} \end{align}$$ All of the nonconstant trajectories are circles of radius $1$, and thus any two points separated by a distance of more than $2$ are not joined by a solution.
If smoothness is an issue, you can replace $\sqrt{\dot{x}^2+\dot{y}^2}$ with $\sqrt{1+\dot{x}^2+\dot{y}^2}$ and get similar behavior.