Hartman–Grobman theorem

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in the study of <a href="/facts/Dynamical_systems/782gREl5">dynamical systems</a>, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of a <a href="/facts/Hyperbolic_equilibrium_point/b31Wvs5W">hyperbolic equilibrium point</a>.  It asserts that <a href="/facts/Linearization/ohYSeQR3">linearisation</a>—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to <a href="/facts/Philip_Hartman/hHVRqRmQ">Philip Hartman</a> and David M. Grobman.
</p><p>The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its <a href="/facts/Linearization/ohYSeQR3">linearization</a> near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria.
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Hartman–Grobman theorem open-in-new

Hartman–Grobman theorem