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The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations. [ABSTRACT FROM AUTHOR]

Vortex-induced vibrations have been recently employed to capture scalable flow energy harvesters, which can attain the maximum power when the wind speed is in the lock-in region where the vortex-shedding frequency is close to the structural frequency. Nevertheless, the dynamical mechanism of the synchronization phenomenon has not been reported. To solve this critical problem, in this paper we explore a novel internal resonance to scavenge flow energy from vortex-induced vibrations, the mechanism of synchronous oscillations is introduced by the amplitude-frequency relationship and confirmed by the energy function. To show the capturing performance, an electromagnetic energy harvester with one-to-one internal resonance is proposed. Based on the harmonic balance method, the electromechanical coupling equations are decoupled, and the first order approximate harmonic responses of displacement and current are established. The modulation equations are derived, the amplitude-frequency curves of displacement and current are plotted with different detuning parameters. The advantage of the proposed one-to-one internal resonance is compared to the noninternal resonance case, the results express that the internal resonance scheme can enhance flow energy capture. The effects of physical parameters on the scavenged power are discussed. The accuracy and efficiency of the approximate analytical results are checked by numerical simulations. [ABSTRACT FROM AUTHOR]