Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour of a linearlized system. One of these two cases is when the linearlized system has a repeated nonzero real eigenvalue; then the equilibrium solution
of the linearlized system is a node, while the original almost linear system could be a node or a spiral, and <b>I wish to see an explicit example of when the spiral occurs.</b> 

To give you an idea of what I need, let's discuss the other case when the linearized system is a center, but the almost linear system could be a center or a spiral, and the example when the spiral occurs is $x^\prime=y+x(x^2+y^2)$ and $y^\prime=-x+y(x^2+y^2)$.