# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## 4 Numerical Results  
The complete unfolding of the codimension-four bifurcation is out of the scope of this paper. Here, however, the turbulence amplitude is kept fixed to the value \(\hat{u}=0.1\), whereas, for different values of the internal detuning \(\sigma_{i}\), the critical and post-critical behavior is analyzed on the plane \((\sigma_{e},\bar{U})\). More specifically, the values \(\sigma_{i}=0\), i.e., perfect internal resonance, and \(\sigma_{i}=-0.06\), i.e., slightly far from the internal resonance, are alternatively considered, in order to assess the transition from a resonant to non-resonant case [50]. The other system parameters assume the following values: \(\kappa=6.5\), \(\eta=0.00026\), \(c_{e}=0.025\), \(b_{1}=-0.046\), \(b_{3}=0.384\), \(\zeta_{1}=0.02\). Different configurations of the viscous-elastic device are also addressed, namely the case of linear controlled beam (LC), i.e., \(\zeta_{3}=0\), to be compared to the outcomes of the nonlinear controlled system (NLC), where \(\zeta_{3}=50\). A further comparison with the uncontrolled (UC) case, realized when the base is fixed (\(\kappa\to+\infty\)), is carried out as well. In the above mentioned cases, geometric and inertia nonlinear contributions are not considered (\(\bar{d}_{j}\) set to zero), in order to be consistent with the model proposed in [37], where nonlinearities come from aerodynamic and viscous device only and where, however, exclusively the first mode was considered (NLC-1M). Finally, the effect of structural geometric and inertial nonlinear contributions is addressed (NLC-NL), through the modification produced by the coefficients \(\bar{d}_{j}\) to the NLC case. The considered cases are summarized in Table 2.  
First, the stability domain of the trivial solution \((a_{1},a_{2})=(0,0)\) is shown in Figure 4 for the uncontrolled and controlled cases, where only the first bifurcations are reported. The plots are in good agreement with those reported in [37] since, due to the nature of the considered damping, always the first mode is directly involved in the bifurcation, being the second one pulled along (as it will be confirmed later). For the same reason, the domains turn out to be insensitive to \(\sigma_{i}\), however, in dependence on \(\sigma_{e}\), the first bifurcation is of Hopf type (H, for \(|\sigma_{e}|<0.0022\)) or of Neimark type (N, for \(|\sigma_{e}|\geq 0.0022\)). For \(|\sigma_{e}|<0.0022\), second Hopf bifurcations are found as well (SH, upper semi-perimeter of the ellipses). The control through the viscous-elastic device (black lines) generally induces an increasing of the critical values of \(\bar{U}\), as expected, whereas the nonlinear terms in the equations do not affect the domains, they being ruled only by the linear ones. In Figure 5, the bifurcation diagrams for amplitudes \(a_{1},a_{2}\) as functions of \(\bar{U}\) are shown, obtained as vertical path in Figure 4 for \(\sigma_{e}=0\). In particular, solid and dashed lines represent stable and unstable equilibria, respectively, whereas the filled regions indicate limit cycles in the amplitudes \(a_{1},a_{2}\), the former corresponding to (stable and unstable) periodic motions of the beam and the latter to quasi-periodic evolution. Comparison between controlled and uncontrolled (grey line, UC) cases shows that the control system pushes forward the first and second Hopf bifurcation points. Specifically, from the first point a stable branch arises, whereas from the second one the branch is unstable. Moreover, the NLC case (red line) shows smaller amplitudes in the post-critical evolution than the LC case (black line). If only one mode is considered (NLC-1M, blue line), a similar behavior to NLC is found for \(a_{1}\). Considering the nonlinear geometric and inertia terms (NLC-NL, orange line) modifies very