the following system of nonlinear equations:

\[eq:mvtR\] $$\ddot{x}(t) + \lambda_0 \dot{x}(t) + \omega_0^2 x(t) + \epsilon_0\left(x(t) - v(t)\right) = f(t)$$ $$\ddot{v}(t) + \lambda_1 \dot{v}(t) + r(\dot v(t),t) + \epsilon_1 \left(v(t)-x(t)\right) = 0$$

where $x$ and $v$ are respectively the displacements of the main mass and of the absorber, $\lambda_0$ and $\lambda_1$ are damping ratios, $\omega_0$ is the natural angular frequency of the linear (uncoupled) oscillator, $f(t)$ represents an external forcing and $\epsilon_0$ and $\epsilon_1$ are coupling ratios such that: $$\frac{\epsilon_0}{\epsilon_1} = \frac{m}{M}$$

The term $r$ represents a nonlinear and hysteretic restoring force which means that it depends of the history of the non-linear motion.

Various kind of systems experience hysteretic behaviours in dynamics; some systems inelastic or with memory may have a restoring force dependent of the history of the deformation, some other, such as rubber or cable isolator are design to dissipate the vibratory energy in the hysteretic loop. The Bouc-Wen differential model, originally proposed by Bouc [@Bouc-Hyst] and reviewed by Wen [@Wen-Hyst], is one of the most used phenomenological model of hysteresis in mechanics. It is also used for system identification of hysteretic systems. The hysteretic force $r(t)$ is based on the displacement $v(t)$ time history and is 