1-In statistics: asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size n may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of n → ∞.

2-In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation

{\displaystyle F(x,y,y',\ \dots ,\ y^{(n-1)})=y^{(n)}\quad x\in [0,+\infty )}

is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution. The number of roots carries also information on the spectrum of associated boundary value problems.

Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional Schrödinger equation the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.

3-In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental.

4-Stability theory was started by Morley (1965), who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by Shelah (1969), who is responsible for much of the development of stability theory. The definitive reference for stability theory is (Shelah 1990), though it is notoriously hard even for experts to read, as mentioned, e.g., in (Grossberg, Iovino & Lessmann 2002, p. 542).

it is a very appreciated and cleared answer: Thank you so much Dr.Maged Gumaan Bin-Saad

All of these are concerns of Dynamical Systems evolving in time, determining the behavior as $t \to \infty$. One may ask whether the system solutions (always) converge to a steady state or oscillate in a bounded (compact?) attracting set.

More generally, Asymptotic Theory may consider approximations as some other parameters tend to limits as, e.g., in (Singular) Perturbations Theory -- often representing the availability of a "good enough" simpler model. Inequalities Theory

then seeks explicit estimates to demonstrate these behaviors.

The asymptotic theory also can be related to the analysis of formal series (not necessarily convergent) related to functions which has the property that truncating the formal series after a finite number of terms provides an approximation to a given function, defined in a sector S, as the argument of the function tends to a point p, such that S has vertex in p.

There are many applications of this theory, especially in Differential Equations and in Quantum Mechanics (Perturbation theory).

Asymptotic theory is a very active research area.

Some references:

Olver, F. (1997). Asymptotics and Special functions. AK Peters/CRC Press.

Wolfgang Wasow (1965) Asymptotic Expansions for Ordinary Differential Equations, Dover/N.Y.

Ovidiu Costin , Asymptotics and Borel Summability (Monographs and Surveys in Pure and Applied Mathematics).

Thanks more prof Fernando Pereira Paulucio Reis for give this answer and references