How to differentiate or infer the equilibrium point from a particular solution of a homogeneous SDE?

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for instance the particular solution of a homogeneous SDE is given by , enter image description here

It looks like a spiral to me. But I cant explain why this is not a centre? Since centre and spirals contain sine and cosine parts.

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Show that $y_1(t)^2+y_2(t)^2=13$ for all $t$. This shows that each pair $(y_1(t),y_2(t))$ lies on the circle $C=\{(x,y) \in \mathbb R^2: x^2+y^2=13\}.$