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rigorously that the dynamics of system (1.6) is Li-Yorke chaotic. In other words, we will show that the chaos of (1.2) is persistent even if it is perturbed with the solutions of system (1.1). 

Sufficient conditions on systems (1.1), (1.2) and (1.6) for the persistence of chaos, and the descriptions concerning almost periodicity and Li-Yorke chaos are provided in the next section. 

## 2 Preliminaries 

Throughout the paper, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices [30]. 

In our theoretical discussions, we will make use of the concept of Li-Yorke chaotic set of functions [19, 31, 32]. The description of the concept is as follows. 

Suppose that Γ ⊂ R p is a bounded set, and denote by 

a collection of uniformly bounded functions. 

<img src='content_image/325822.jpg'>

A couple of functions ( ψ ( t ) , ψ ( t ) ∈ H×H is called proximal if for arbitrary small ϵ > 0 and arbitrary large E > 0 , there exists an interval J with a length no less than E such that ∥ ψ ( t ) − ψ ( t ) ∥ < ϵ for each t ∈ J . On the other hand, we say that the couple ( ψ ( t ) , ψ ( t ) ∈ H × H is frequently ( ϵ$_{0}$, Δ) − separated if there exist positive numbers ϵ$_{0}$, Δ and infinitely many disjoint intervals with lengths no less than Δ such that ∥ ψ ( t ) − ψ ( t ) ∥ > ϵ$_{0}$ for each t from these intervals. We call the couple ( ψ ( t ) , ψ ( t ) ∈ H × H a Li-Yorke pair if it is proximal and frequently ( ϵ$_{0}$, Δ) -separated for some positive numbers ϵ$_{0}$ and Δ . Moreover, a set S ⊂ H is called a scrambled set if S does not contain any almost periodic function and each couple of different functions inside S × S is a Li-Yorke pair. 

H is called a Li-Yorke chaotic set if: ( i ) H admits a countably infinite subset of almost periodic functions; ( ii ) there exists an uncountable scrambled set S ⊂ H ; ( iii ) for any function ψ ( t ) ∈ S and any almost periodic function ψ ( t ) ∈ H , the pair ( ψ ( t ) , ψ ( t ) is frequently ( ϵ$_{0}$, Δ) − separated for some positive numbers ϵ$_{0}$ and Δ . 

Remark 2.1 The criterion for the existence of a countably infinite subset of almost periodic functions in a Li-Yorke chaotic set can be replaced with the existence of a countably infinite subset of quasi-periodic or periodic functions. 

The presence of Li-Yorke chaos with a basis consisting of infinitely many almost periodic motions in the dynamics of system (1.1) is one of our main assumptions. In other words, we suppose that system (1.1) possesses a set A of uniformly bounded solutions which is chaotic in the Li-Yorke sense. In this 

