The Hartman-Grobman Theorem

Fractional dynamical systems: A fresh view on the local qualitative theorems

The aim of this work is to describe the qualitative behavior of the solution set of a given system of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. In order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. This is done by the extension of...

The Hartman-grobman Theorem and the Equivalence of Linear Systems

An Undergraduate’s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems

The Hartman-Grobman and Poincaré-Bendixon Theorems are two of the most powerful tools used in dynamical systems. The Hartman-Grobman theorem allows us to represent the local phase portrait about certain types of equilibria in a nonlinear system by a similar, simpler linear system that we can find by computing the system’s Jacobian matrix at the equilibrium point. The Poincaré-Bendixon theorem g...

Lyapunov function proof of Poincaré's theorem

One of the most fundamental results in analysing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincaré’s theorem. The proof of this result involves system analytic arguments along with the Hartman–Grobman theorem. Using the notions of stability of sets, lower semicontinuous Lyapunov functions are constructed to provide a Lyapunov function proof of Poincar...

On the Grobman-hartman Theorem in Α-hölder Class for Banach Spaces

We consider a hyperbolic diffeomorphism in a Banach space with a hyperbolic fixed point 0 and a linear part Λ. We define σ(Λ) ∈ (0, 1], and prove that for any α < σ(Λ) the diffeomorphism admits local α-Hölder linearization.