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The paper develops and studies a very general notion of dichotomy, referred to as “nonuniform (h,k,µ,ν)-dichotomy”. The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then provides corresponding new versions of robustness, Hartman-Grobman theorem, and stable manifold theorem for nonautonomous dynamical systems in Banach spaces in term of the nonuniform (h,k,µ,ν)-dichotomy.

A dynamical system admitting an invariant manifold can be interpreted as a single element of an infinite class of dynamical systems that all exhibit the same behaviour on the invariant manifold. This observation is used in the context of autonomous ordinary differential equations to generalize a global stability result of Li and Muldowney. The new result is demonstrated on an epidemiological model.

In this paper we show that the set of solutions of the Nicoletti or Floquet boundary value problems for hyperbolic differential equations is nonempty compact and convex. We apply the Browder–Godhe–Kirk fixed point theorem.

In applications of the Internal Model Principle of control theory to nonlinear differential systems, a semistable-center-unstable invariant manifold plays an important role. The existence of this manifold has been known for some time: in this paper its smoothness is related to the location of the spectrum of the linearization of the system at the critical element.

In this paper, we consider a general autonomous functional differential equation having a local center manifold. Then, we give an algorithmic procedure to compute the terms in the Taylor expansion of this manifold up to any order.