Dynamics of a Tree Trunk Structure in Double-Well Potential under Dual Turbulent Flow and Periodic Loading

This study explores the nonlinear dynamics of a tree trunk subjected to combined turbulent and periodic excitations. The mathematical modeling yields a Duffing oscillator driven by a constant force, an amplitude-modulated force and a harmonic force. Analysis of the vibrational resonance reveals the existence of both single and double resonance peaks, whose amplitudes are significantly attenuated by the damping coefficient. A comparison between theoretical and numerical approaches shows excellent agreement, thereby validating the method of direct separation of motions employed. The transition to chaos is investigated using Melnikov’s method, which allows for the analytical definition of threshold curves for horseshoe chaos, highlighting the critical influence of the periodic excitation frequency. Numerically, the fourth-order Runge–Kutta algorithm reveals a rich variety of dynamical behaviors, including (both forward and reverse) period-doubling cascades, anti-monotonicity, symmetry-breaking, chaos and periodic/chaotic bubbles. Crucially, it is observed that when the damping coefficient is employed as a bifurcation parameter, its increase systematically suppresses chaos, leading the system toward stable periodic behavior. This identifies the damping capacity as a vital safety mechanism for structural integrity. Finally, the system exhibits pronounced multistability, characterized by the coexistence of periodic and chaotic attractors of various topologies, confirming the structure’s extreme sensitivity to initial conditions and the complexity of its mechanical survival.