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Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass, a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra, and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed. The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.

Linear and weakly nonlinear analyses are made for the Rayleigh-B´enard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret co-efficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation. Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The in-fluence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of ap-pearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.

This paper presents a novel mechanical attachment, i.e., nonlinear energy sink (NES), for suppressing the limit cycle oscillation (LCO) of an airfoil. The dynamic responses of a two-degree-of-freedom (2-DOF) airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.

In this paper, a Duffing oscillator model with delayed velocity feedback is considered. Applying the time delayed feedback control method and delayed differential equation theory, we establish some criteria which ensure the stability and the existence of Hopf bifurcation of the model. By choosing the delay as bifurcation parameter and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. We found that if the time delay is chosen as a bifurcation parameter, Hopf bifurcation occurs when the time delay is changed through a series of critical values. Some numerical simulations show that the designed feedback controllers not only delay the onset of Hopf bifurcation, but also enlarge the stability region for the model.

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable＇s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable＇s triad resonant re- sponses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable＇s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable＇s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the criti- cal amplitude of high-frequency mode to activate cable＇s triad resonance, and the second characterizes the degree of difficulty for activating cable＇ s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.

Acoustical signal transduction in the cochlea is an active process that involves nonlinear amplification and spontaneous otoacoustic emissions. Signal transduction involves individual subunits composed of globally coupled hair cells, which can be modeled as oscillators close to a Hopf bifurcation. The coupling may induce a transition toward synchronization, which in turn leads to a strong nonlinear response. In the model studied here, the synchronization transition of the subunit is discontinuous (explosive) in the absence of an external stimulus. We show that, in the presence of an external stimulus and for a coupling strength slightly lower than the critical value leading to explosive synchronization, the response of the subunit has better frequency selectivity and a larger signal-to-noise ratio. From physiological observations that subunits are themselves coupled together, we further propose a model of the complete cochlea, accounting for the ensemble of frequencies that the organ is able to detect.

The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit.Local active coordinate system is introduced to construct the Poincar′e returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the perioddoubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2n-homoclinic orbits.

The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.

This paper discuss the cusp bifurcation of codimension 2 (i.e. Bogdanov-Takens bifurcation) in a Leslie-Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14(1) (2010), 289–306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.