## 1 Introduction  
In the last decades, great interest has been given to the dynamic phenomena typical of nonlinear systems in view of exploiting them to mitigate the effects of dynamic excitations [1, 2, 3, 4]. One of the usual techniques used to reduce structural vibration is to introduce elements that guarantee a higher dissipation capacity [5]; here, the choice of introducing hysteretic elements allows the combination of two strategies, which increase the ability of the structure to dissipate energy and, at the same time, exploit the beneficial aspects of the nonlinear dynamic coupling caused by the hysteresis itself [6, 7, 8, 9]. The variety of the nonlinear phenomena is very large [10]; in particular, the modification of the modal quantities, with the oscillation amplitude, assumes great importance and entails change in the natural frequencies and in the linear modes. They are no longer orthogonal, and in a periodic cycle, they change their shape, leading to modal trajectories that are no longer straight in the configuration plane [11, 12]. The occurrence of internal resonance conditions is easily met, and in these conditions, the modal coupling is exalted [4, 7]. Several nonlinear motions can take place, such as multifrequencies or quasiperiodic and chaotic oscillations [10, 13, 14]; however, the attention here is focused on more robust phenomena related to periodic motion and features related to cases of strong nonlinearities. Under these conditions, an unusual phenomenon occurs when a novel mode arises with respect to the linear ones, as already observed in [7, 11, 15, 16]. Hysteretic behavior is widespread among materials and elements; it is easily encountered in civil, mechanical, aeronautical and electrical engineering, and it is characterized by an intrinsic capacity for energy dissipation [6, 7, 8, 9, 17, 18]. Some examples are materialswith limited elastic fields, elastomeric absorbers, shape-memory alloys, contacts exhibiting microsliding friction and magnetostrictive materials. Compared to geometric nonlinearities, those introduced by hysteresis can produce more marked effects on the dynamic response [14; 17; 18; 19; 20; 21; 22] and give rise to phenomena typical of nonlinear dynamics, such as modal coupling, bifurcations and the emergence of superabundant modes [12; 16; 23; 24]. A system is said to be endowed with hysteresis if the output depends, in a rate-independent way, on the history of the input [25]; however, this definition does not consider that rate-dependent hysteresis can be encountered in a number of applications involving smart materials, piezoelectric actuators or other specific devices [26]. According to this definition, the main characteristic of hysteresis is thus its dependence on the history of the input; while rate independence relatively eases the treatment of the problem, the dependence on the history makes it very difficult to obtain analytically tractable formulations. Thus, beyond the high number of different restoring forces encountered in the real world, this explains the high number of models proposed in the literature, even when limited to the mechanical field. Among the differential models, the Bouc-Wen model [27; 28] was adopted because it is very versatile in describing various hysteretic behaviors and simple due to its ability to give analytical piecewise smooth representations of stress-strain relationships. The versatility of the model in describing restoring forces with different characteristics has been demonstrated in several papers [14; 29; 30; 31]; however, here, the Bouc-Wen law was adopted in its basic formulation because the interest is mainly devoted to exploiting the use of generic hysteretic behaviors for vibration mitigation. For this aim, a 2 DOF system with one hysteretic element, representative of a multidegree-of-freedom system under harmonic excitation, was investigated.