User:Hyiltiz

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.