(Solved): The system of non-linear differential equations \[ \begin{array}{l} \dot{x}=\sin y \cos (\pi x), \ ...

The system of non-linear differential equations \[ \begin{array}{l} \dot{x}=\sin y \cos (\pi x), \\ \dot{y}=\sin y+\cos \left(\frac{\pi x}{2}\right), \end{array} \] has an equilibrium point at \( (1,0) \). (a) Calculate the Jacobian matrix of this system of equations and evaluate this matrix at the given equilibrium point. (b) Use your answer to part (a) to classify this equilibrium point.