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What is the behaviour of the steady state $(0,0)$ of the system $$\left\{ \begin{array}{c} x' = x + x \cdot \sin{y} \\\ y' = 2y - y \cdot \cos{x} \end{array}\right.$$

Can anybody help me with the solution of this question.

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up vote 1 down vote accepted

Well, seems to be a homework. First even though you don't ask about it, I would mention that there is indeed only one equilibrium because $y' = y(2-\cos x)$.

The first step would be to linearize the system, i.e. for $$\begin{cases} f(x,y) = x(1+\sin y)\\g(x,y) = y(2-\cos x) \end{cases} $$ to find the matrix of partial derivatives $$ A = \left(\begin{align}&f_x(0,0)& f_y(0,0)\\&g_x(0,0) &g_y(0,0)\end{align}\right) $$ and find eigenvalues of $A$.

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