Non-linear dynamics of the oscillations of the plant in a vegetation cover situation under the effects of the wind

This work considers the nonlinear dynamics of the oscillations of plants in situations of vegetation cover under the effect of the wind. A row of identical oscillators is considered to take into account the contribution of close neighbors on the dynamics of an individual plant and the beam theory is used to model the equation of motion of the system. The method of multiple scales is used to obtain the amplitudes of plants under the excitation of the wind in the cases of resonant and nonresonant oscillations. For the resonant case, the harmonic, superharmonic and subharmonic oscillations are studied and it is observed that the crown spring parameter and the nonlinearity coefficients of the third and fifth order influence qualitatively and quantitatively the response of the system. For both types of nonlinearities, the resonant behavior was pushed to higher frequencies. However, the comparative study of the effects of the two types of nonlinearities on the system suggested that the amplitude of the plant oscillations increases with the decreasing resonance frequency. The crown spring parameter had the effect of decreasing the amplitudes and reducing the effect of the nonlinearities. These results prove that the contribution of the close neighbors and the nonlinearities may serve to stabilise plants against lodging and windthrow. Furthermore, using the fourth-order Runge–Kutta algorithm, we have numerically solved the basic equations of the system and very interesting results are obtained. The influences of the crown spring parameter and the fifth-order nonlinearity on the chaotic dynamics of the plant are analyzed. It appears that the bifurcation curves and the Lyapunov exponent are in agreement with the phase diagrams and confirm the analytical results of the system. Graphical abstract