The main goal of the present study is the verification of the persistence of chaos in unidirectionally coupled Lorenz systems even if they are not synchronized in the generalized sense. We rigorously prove that sensitivity is a permanent feature of the response system, and we numerically demonstrate the existence of unstable periodic orbits embedded in the chaotic attractor of the response benefiting from a period-doubling cascade [13] of the drive. Conditional Lyapunov exponents [9] and auxiliary system approach [12] are utilized to show the absence of GS. Our results reveal that the chaos of the drive system does not annihilate the chaos of the response, i.e., the response remains to be unpredictable under the applied perturbation. 

The usage of exogenous perturbations to generate chaos in coupled systems was proposed in the studies [14]-[20]. In particular, the paper [19] was concerned with the extension of sensitivity and periodic motions in unidirectionally coupled Lorenz systems in which the response system is initially non-chaotic, i.e., it either admits an asymptotically stable equilibrium or an orbitally stable periodic orbit in the absence of the driving. However, in the present study, we investigate the dynamics of coupled Lorenz systems in which the response system is chaotic in the absence of the driving. 

Another issue that was considered in [19] is the global unpredictable behavior of the weather dynamics. We made an effort in [19] to answer the question why the weather is unpredictable at each point of the Earth on the basis of Lorenz systems. This subject was discussed by assuming that the whole atmosphere of the Earth is partitioned in a finite number of subregions such that in each of them the dynamics of the weather is governed by the Lorenz system with certain coefficients. It was further assumed that there are subregions for which the corresponding Lorenz systems admit chaos with the main ingredient as sensitivity, which means unpredictability of weather in the meteorological sense, and there are subregions in which the Lorenz systems are non-chaotic. It was demonstrated in [19] that if a subregion with a chaotic dynamics influences another one with a non-chaotic dynamics, then the latter also becomes unpredictable. The present study takes the results obtained in [19] a step further such that the interaction of two subregions whose dynamics are both governed by chaotic Lorenz systems lead to the persistence of unpredictability. 

The rest of the paper is organized as follows. In Section 2, the model of coupled Lorenz systems is introduced. Section 3 is devoted to the theoretical discussion of the sensitivity feature in the response system. Section 4, on the other hand, is concerned with the numerical analyses of coupled Lorenz systems for the persistence of chaos as well as the absence of GS. The existence of unstable periodic motions embedded in the chaotic attractor of the response is demonstrated in Section 5. Some concluding remarks are given in Section 6, and finally, the proof of the main theorem concerning sensitivity is provided in the Appendix. 

