Given that state space trajectories of an autonomous system do not cross, can I deduce that a mapping function f:(x,y)→(x',y') given by a solution of an ODE of an autonomous system is bijective?
Other ideas on how to test if such a system is bijective will be appreciated.
note: the ODE set is non-linear and coupled so there is no explicit term for the ODE solution.
If you have an ODE $$\dot{x} = f(x), \quad x(0) = x_0$$ Denote families of solutions as $\{g(\cdot \ ;x_0)\}$, and you know that the trajectories do not cross on some set $[0,T]\times U$, with $U\subset \mathbb{R}^n$, then the flow map $$\phi^t :U \mapsto \phi^t(U)$$ $$x_0 \mapsto g(t;x_0)$$ Will be bijective for all $t\in [0,T]$. This is what is meant by the trajectories do not cross. Otherwise $g(t;x_0) = g(t;y_0)$, so the only way for this to happen is if the initial data is the same $x_0 = y_0$.