User:Hyiltiz

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State u(t) == R^t

Main theorem

Consider a system evolving in time with state $R^{t}\in \mathbb {R} ^{n}$ that satisfies the differential equation $dR^{t}/dt=g(R^{t})$ for some smooth map $g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$. Suppose the map has a hyperbolic equilibrium state $R^{*}\in \mathbb {R} ^{n}$: that is, $g(R^{*})=0$ and the Jacobian matrix $A=[\partial g_{i}/\partial r_{j}]$ of $g$ at state $R^{*}$ has no eigenvalue with real part equal to zero. Then there exists a neighbourhood $N$ of the equilibrium $R^{*}$ and a homeomorphism $h:N\to \mathbb {R} ^{n}$, such that $h(R^{*})=0$ and such that in the neighbourhood $N$ the flow of $dR^{t}/dt=f(R^{t})$ is topologically conjugate by the continuous map $U=h(R^{t})$ to the flow of its linearisation $dU/dt=AU$.

Even for infinitely differentiable maps $f$, the homeomorphism $h$ need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with an exponent depending on the constant of hyperbolicity of $A$.

The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems $du/dt=f(u,t)$ (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.