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be explained mathematically by showing the existence of a transverse homoclinic point of the time map with the corresponding invariant Smale horseshoe [28]. In general, however, it is not easy to show the existence of a transverse homoclinic point. To this purpose, the perturbation approach, by now known as the Melnikov method, is a powerful method [22]-[24]. The starting point is a nonautonomous system, the unperturbed system/equation, with a (necessarily) nontransverse homoclinic orbit. Then it is known that, if we take the perturbed system by adding a periodic (or almost periodic) perturbation of sufficiently small amplitude to the unperturbed system and a certain Melnikov function has a simple zero at some point, the perturbed system has a transverse homoclinic point with the corresponding Smale horseshoe [25]. 

In the present study, we will consider these systems and perturb them in a unidirectional way through exogenous chaotic forcing terms to achieve propagation of the chaotic behavior. In other words, the influence of the chaos of a system on another one will be mentioned in the paper such that as a result the latter behaves also chaotically. Chaotification of systems with asymptotically stable equilibriums through different type of perturbations can be found in [1, 3, 4, 5, 7, 8, 9]. Endogenously generated chaotic behavior of systems are well investigated in the literature. The systems of Lorenz [36], Rössler [44] and Chua [17, 18] as well as the Van der Pol [16, 32, 33] and Duffing [37, 38, 47] oscillators can be considered as systems which are capable of generating chaos endogenously. We will make use of Li- Yorke chaos in the extension mechanism through exogenous perturbations, and essentially indicate in the present paper that not only endogenous structure of systems, but also exogenous chaotic perturbations can give rise to observation of chaotic behavior. 

Infinitely many periodic solutions can serve as a basis of Li-Yorke chaos as well as infinitely many quasi-periodic or almost periodic solutions can also serve as a basis for the developed Li-Yorke chaos [10]. 

In the present study, we follow the definition of continuous Li-Yorke chaos modified in [10] and consider quasi-periodic and almost periodic solutions as a basis of Li-Yorke chaos. 

In the next section, we introduce the systems which will be under investigation and give information about the properties of these systems under some conditions. 

## 2 Preliminaries 

Let us consider the systems 

and 

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