The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields that dictate the evolution of states).
But is this theorem necessary for this to be true? Can a linearized system with a single eigenvalue on the imaginary axis not behave like its nonlinear counterpart?
I would request any specific references to the given answers so that I can read it and verify by myself!
As @Evgeny commented, the system $\dot x = x^2$ is one of the simplest examples. The situation is analogical to trying to determine whether a local extremum of a function is maximum or minimum when the second derivative is zero: E.g. $x^3$ and $x^4$ have booth first and second derivative zero at $x=0$, but only the later has local minimum at the point.
I recommend the book by Phase Portraits of Planar Quadratic Systems 2007 by John Reyn to see endless examples demonstrating why the condition that the real part of each of the eigenvalues is nonzero is essential.