A paradigmatic system to study the transition from zero/Hopf to double-zero/Hopf bifurcation

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

“…In particular, the effect of the NES on the response of the main oscillator and the consequent energy exchange under primary and subharmonic resonances [34] is investigated. Following what was first done in [35] analyzing the more general doublezero/Hopf bifurcation phenomenon, and then specializing the problem to system with NES devices in [9,13,20,21], the Multiple Scale/Harmonic Balance Method (MSHBM) is used to obtain amplitude modulation equations and to study the dynamic evolution of the system in the slow time scale, allowing one to apply the harmonic balance just to the NES equation as well as retrieving higher frequency contributions through the higher order perturbation solution of the main structure. The response is analyzed in three different cases: (a) the excitation induces exclusively 1:1 resonance, i.e.…”

Galloping of internally resonant towers subjected to turbulent wind

“…(13b): under those conditions, the obtained AMEs would be the same as the corresponding equations for non-resonant case, already discussed in [38]. Therefore, as in [21], the equations ruling the resonant case (with higher codimension) tend to those applying to the non-resonant case, and they can be used from small to large values of the parameter.…”

“…The bifurcation analysis is carried out in planar sections of the multiple-dimensional bifurcation parameter space, and the equilibrium solutions of amplitude modulation equations (AMEs), representing periodic or quasi-periodic oscillations of the towers, are analyzed. Moreover, a discussion of the transition from the internally resonant case to the nonresonant one is done, in the same framework as [21] where changeover from double-zero-Hopf to zero-Hopf bifurcation was analyzed.…”

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

“…In particular, the topics that can be recognized in these papers concern the investigation of: (a) divergenceHopf bifurcation [56][57][58], (b) double-zero and multiple-zero bifurcations [59,60], (c) simple Hopf bifurcation [61], multiple (non-defective) Hopf bifurcation [62,63] and multiple (defective) Hopf bifurcation, [64], (d) theoretical and technical [65,66], and algorithmic [67] aspects in perturbation methods.…”

“…3. In [58], a nonstandard application of the MSM, devoted to analyze the transition from codimension-3 (double-zero/Hopf) to codimension-2 (single-zero/Hopf) bifurcations, occurring in a two DOF, is presented. The bifurcation equations of the system, here obtained, lead to a four-dimensional dynamical system, consisting of a first-order complex equation (as in the Hopf bifurcation) and a second-order real equation (as in the Takens-Bogdanov bifurcation), coupled by mixed terms; it is shown that this system can be reduced to a three-dimensional system, coherently with the codimension of the problem.…”