In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

The paper presents some characterizations for the study of dichotomy of dynamical systems. Integral characterizations of Datko type are considered for the notion of uniform dichotomy with growth rates, also called uniform h-dichotomy, for skew-evolution cocycles in Banach spaces.

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented.

The interdependency between interest rates, investment demands and inflation rates in a given economy has a continuous dynamics. We propose a four-dimensional model which describes these interactions by imposing a control law on the interest rate. By a qualitative analysis based on tools from dynamical systems theory, we obtain in the new model that the three economic indicators can be stabilized to three equilibrium states.

In this paper is considered the concept of uniform dichotomy with differentiable growth rates for skew-evolution cocycles in Banach spaces which includes as particular case, the well-known concept of uniform exponential dichotomy. Necessary and sufficient conditions are presented using invariant families of projectors.

This paper introduces a new approach to integral characterizations in the study of dynamical systems, utilizing the connection between uniform dichotomy with differentiable growth rates and uniform exponential dichotomy. Within this framework, we establish integral conditions for these dichotomies, considering both invariant projection-valued functions and projection-valued functions compatible with a skew-evolution cocycle. The method relies on incorporating differentiable growth rates, which extend existing results and provide a broader perspective on dichotomy theory. We hope that these results may contribute to the dichotomy theory of nonautonomous dynamical systems.