## What is/are Exponential Dichotomy?

Exponential Dichotomy - It is closely related to the concept of exponential dichotomy and has played a key role in the theory of Lyapunov exponents.^{[1]}Using the obtained a priori estimates for the corresponding operator, the property of the exponential dichotomy for a generating homogeneous equation is established.

^{[2]}The notion of exponential dichotomy is used to characterize the normal hyperbolicity and a generalized Lyapunov-Perron approach is used in order to prove our main result.

^{[3]}The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis.

^{[4]}On the other hand, our abstract results fit perfectly with the case that the family $$(A(t))_{t\ge 0}$$(A(t))t≥0 generates an evolution family having exponential dichotomy.

^{[5]}In this paper we prove versions, in Frechet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces.

^{[6]}In this work, we study the existence and uniqueness of almost automorphic solutions for semilinear nonautonomous parabolic evolution equations with inhomogeneous boundary conditions using the exponential dichotomy.

^{[7]}For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy.

^{[8]}The working tools are based on the contraction mapping principle, the theory of exponential dichotomy and Lyapunov functions.

^{[9]}There are examples showing that in this case we can't say anything about the exponential dichotomy of this system (that is, its regularity).

^{[10]}In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE.

^{[11]}Moreover, each semigroup (e)τ≥0 is supposed to have exponential dichotomy and the Hölder constant of A(·)−1 must be sufficiently small.

^{[12]}

## fixed point theorem

In this letter, the existence and the global exponential stability of piecewise pseudo almost periodic solutions (PAPT) for bidirectional associative memory neural networks (BAMNNs) with time-varying delay in leakage (or forgetting) terms and impulsive are investigated by applying contraction mapping fixed point theorem, the exponential dichotomy of linear differential equations and differential inequality techniques.^{[1]}Based on the exponential dichotomy of linear differential equations, the Banach fixed point theorem and the differential inequality technique, we obtain the existence of almost periodic solutions of this class of networks.

^{[2]}By applying the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques, we obtain some sufficient conditions which guarantee the existence and exponential stability of almost periodic solutions for such class of BAM neural networks.

^{[3]}To do so, the exponential dichotomy theory, the contraction mapping fixed point theorem and the differential inequality techniques are used.

^{[4]}

## Nonuniform Exponential Dichotomy

The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.^{[1]}Under the condition of nonuniformly bounded growth, %nonuniform exponential dichotomy spectrum for nonautonomous linear system is proposed the relationship of the nonuniform exponential dichotomy spectrum and the other two classical spectrums (the Lyapunov spectrum and Sacker-Sell spectrum) is given, and the stability of these spectrums under small linear perturbations are summarized and presented in this paper.

^{[2]}In this paper, we discuss the nonuniform exponential dichotomy properties of nonautonomous systems of linear differential equations.

^{[3]}The latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part, which is ubiquitous in the context of ergodic theory.

^{[4]}

## Strong Exponential Dichotomy

In this paper we investigate $$C^1$$ smooth linearization for nonautonomous difference equations with a nonuniform strong exponential dichotomy.^{[1]}In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy.

^{[2]}

## exponential dichotomy property

Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy properties in terms of semigroup theory.^{[1]}In this paper, we discuss the nonuniform exponential dichotomy properties of nonautonomous systems of linear differential equations.

^{[2]}