Consider the stability of the equilibrium point of the system $$ \dot{x}=\mathrm{e}^{x+y}-\sqrt{1+2x} $$ $$ \dot{y}=\sin(xy)+\ln(1+2x) $$
The equilibrium of the system is $(0,0)$. My problem is to determine whether it is Lyapunov stable/asymptotically stable or not. I am currently working on using contraction. To be more precise, if we can prove for some positive initial value, $x(t), y(t)$ are all positive, then $\dot{y}$ must have a positive lower bound, but I cannot write a clear proof. If my way is wrong, please suggest your method. By the way, is there any more generalized criterion suitable for the problem?
Appreciate any help!