A global Hartman-Grobman theorem (Q2155952)

In this article, the authors prove two generalizations of the Hartman-Grobman theorem. In the first result, considering the nonlinear system \[ \dot{x}=f(x), \] with \(f:E\mapsto\mathbb{R}^{n}\) of class \(C^{1}\) and assuming that the origin is a hyperbolic equilibrium point, then for any bounded set \(A\subset E\) containing the origin, there exists a homeomorphism \(H\) defined in a neighborhood of the origin \(V(0)\subset A\) which transforms the nonlinear system into the linear system \[ \dot{y}=Df(0)y. \] On the other hand, the second result is contextualized in the hypotheses of the first theorem, either by considering that all the eigenvalues of the matrix \(Df(0)\) are located on the left of the imaginary axis or on the right of the imaginary axis, then there exists a homeomorphism defined in the region of attraction/repulsion of the origin of the nonlinear system, and this system is transformed into its linearized system. Finally, different examples are presented in the framework of each result.