where the function F : Λ → Λ is continuous and Λ is a bounded subset of R $^{n}$. Here, R and Z denote the sets of real numbers and integers, respectively. The system under investigation is a hybrid one, since it combines the dynamics of an impulsive differential equation with a discrete map. Our main objective is to prove rigorously the existence of homoclinic and heteroclinic solutions in the dynamics of (1.1) provided that (1.2) possesses such solutions. 

The idea of the usage of discontinuous perturbations to generate homoclinic and heteroclinic motions in systems of differential equations was first realized in the papers [3, 7] on the basis of functional spaces. It was shown in [3] that the chaotic attractor of the relay system, which was introduced in the paper [6], consists of homoclinic solutions. Similar results for impulsive differential equations were obtained in the study [7] by taking advantage of the moments of impulses. 

The existence of homoclinic and heteroclinic motions in systems with impulses were also investigated in the papers [9, 11, 12, 14, 18, 26, 27]. The existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects were investigated in [26] by using the mountain pass theorem and the symmetric mountain pass theorem in the critical point theory. The mountain pass theorem was also utilized in [11, 18] to show the presence of homoclinic motions in second order impulsive systems. On the other hand, Wei and Chen [22, 23] considered the existence of heteroclinic cycles in predator-prey systems with Allee effect and state-dependent impulsive harvesting within the scope of their studies. Zhang and Li [27] proved the existence of at least one non-zero homoclinic solution, which is generated by impulses, under appropriate conditions for a class of impulsive second order differential equations. Han and Zhang [14] obtained the existence of homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulses by using variational methods. It was mentioned in [14] that no homoclinic solutions exist for the system under investigation without impulses. However, in the present study, the emergence of homoclinic and heteroclinic motions are completely provided by the influence of a discrete map instead of impulsive effects. Additionally, our results are valid for systems with arbitrary high dimensions. 

The rest of the paper is organized as follows. In Section 2, we discuss bounded solutions of (1.1), and present sufficient conditions for the existence of homoclinic and heteroclinic motions in the system. Section 3 is devoted for the main results of the paper. In this part, we show the connection between the stable and unstable sets of the impulsive system (1.1) and the discrete map (1.2), and prove the existence of homoclinic and heteroclinic solutions in (1.1). Examples concerning homoclinic and heteroclinic motions in an impulsive Duffing equation are provided in Section 4. Finally, some concluding remarks are given in Section 5. 

