In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behavior of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization—a natural simplification of the system—is effective in predicting qualitative patterns of behavior.
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The theorem states that the behavior of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behavior of its linearization 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 analyze its behavior around equilibria.
Main theorem
Consider a system evolving in time with state
Even for infinitely differentiable maps
The Hartman–Grobman theorem has been extended to infinite dimensional Banach spaces, non-autonomous systems
Example
The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic.
Consider the 2D system in variables
By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is
is a smooth map between the original
That is, a distorted version of the linearization gives the original dynamics in some finite neighbourhood.