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Based on the exponential dichotomy of linear dynamic equations on time scales, we obtain some sufficient conditions for the existence and global exponential stability of almost periodic solutions for a class of Duffing equations with time-varying delays on time scales. We also present numerical examples to show the feasibility of obtained results. The results of this paper are completely new even when the time scale

As we know Duffing equations describe the motion of a mechanical system in a twin-well potential field. Due to their promising potential applications in areas of physics, mechanics, and engineering technique fields, various kinds of dynamic behaviors of Duffing equations have been studied by many authors (see [

However, there have been few results about the discrete analogue of the above systems. In fact, both continuous and discrete systems are very important in implementation and applications. But it is troublesome to study the dynamics for continuous and discrete systems, respectively. Therefore, it is significant to study that on time scales which can unify the continuous and discrete situations.

Motivated by the above, in this paper, we study the almost periodic solutions of the following Duffing equation on time scale

If

Let

To the best of our knowledge, up to now, there are no results available on the existence and global exponential stability of almost periodic solution for Duffing equations on time scales. Our main aim of this paper is to study the existence of almost periodic solutions for (

For convenience, we denote

Throughout this paper, we assume that the following condition holds:

This paper is organized as follows. In Section

In this section, we introduce some definitions and state some preliminary results.

Let

Assume that

Let

if

Assume that

A function

Suppose that

if

If

A time scale

Let

If

If

If

Let

If (

If

Let

In this section, we will state and prove the existence and global exponential stability of almost periodic solutions of (

Let

Let

there exists a positive constant

the following inequalities hold:

For any given

At first, we show that for any

Let

By Theorem

It is easy to see that when

In this section, we present numerical examples to illustrate the feasibility of our results obtained in Section

Consider the delay Duffing equation on an almost periodic time scale

Consider the delay Duffing equation on

Using the exponential dichotomy of linear dynamic equations on time scales and the time scale calculus theory, some sufficient conditions are derived to guarantee the existence and exponential stability of almost periodic solutions for a class of Duffing equation on time scales. To the best of our knowledge, the results presented here have not appeared in the related literature. Besides, the results obtained in this paper possess feasibility. Moreover, the method in this paper may be applied to some other differential equations on time scales.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the referees for their careful reading of the paper, support, and insightful comments. This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.