I would be very grateful if someone clever explained to me why Hartman-Grobman theorem does not work when one of the eigenvalues of linearized system is purely imaginary? Is there any intuition behind this?
2026-05-04 14:53:48.1777906428
Why Hartman-Grobman theorem does not work when one of the eigenvalues is purely imaginary?
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I'll try to answer it but this is something that I 'intuitively' feel like. Consider the linearized system with eigenvalue at the imaginary axis. Consider the neighbourhood of such an eigenvalue (in the complex plane) and you would realize the behavior of the system 'discontinuous in nature'. While on one hand, the neighbourhood to the left attracts the flow lines (corresponding to the vector fields associated with state evolution) while, on the other hand, the right side repels it. Hence, with linearization you would need not one but two (or maybe more?) mathematical functions to represent the behavior of the flow lines without any continuity at the point itself. This causes the homeomorphic transformation to be not continuous in nature (possibly undefined or singular at the point itself?). I would love to hear other's opinion on this!