Experimental study of the Hopf bifurcation in rectangular low-density jets

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.

ABSTRACTMost researches on railway vehicle stability analysis are focused on the codimension 1 (for short, codim 1) bifurcations like subcritical and supercritical Hopf bifurcation. The analysis of codim 1 bifurcation can be completed based on one bifurcation parameter. However, two bifurcation parameters should be considered to give a general view of the motion of the system when it undergoes a degenerate Hopf bifurcation. This kind of bifurcation named the generalised Hopf bifurcation belongs to the codimension 2 (for short, codim 2) bifurcations where two bifurcation parameters need to be taken into consideration. In this paper, we give a numerical analysis of the codim 2 bifurcations of a nonlinear railway wheel-set with the QR algorithm to calculate the eigenvalues of the linearised system incorporating the Golden Cut method and the shooting method to calculate the limit cycles around the Hopf bifurcation points. Here, we found the existence of a generalised Hopf bifurcation where a subcritical Hopf bifurcation turns into a supercritical one with the increase of the bifurcation parameters, which belong to the codim 2 bifurcations, in a nonlinear railway wheel-set model. Only the nonlinear wheel/rail interactive relationship has been taken into consideration in the lateral model that is formulated in this paper. The motion of the wheel-set has been investigated when the bifurcation parameters are perturbed in the neighbourhood of their critical parameters, and the influences of different parameters on critical values of the bifurcation parameters are also given. From the results, it can be seen that the bifurcation types of the wheel-set will change with a variation of the bifurcation parameters in the neighbourhood of their critical values.

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

In this paper, the predator–prey model with Michaelis–Menten functional response subject to the Neumann boundary conditions is considered. The stability of the positive equilibrium and the direction of the Hopf bifurcations for the corresponding ordinary differential equation and partial differential equation are studied, respectively. To this end, the normal form method and the center manifold theorem are applied to determine the direction of the Hopf bifurcation and the stability of the bifurcated limit cycle when the diffusion terms are present. It is shown that the corresponding system can undergo either the supercritical or subcritical Hopf bifurcation at the equilibrium point within certain parameter range. Meanwhile, the Turing instability conditions for the equilibrium and the limit cycle bifurcated from the Hopf bifurcation are derived. As a result, some patterns occur in the model. Numerical simulations are carried out to verify the theoretical analysis.

Uniaxial strength of a composite array of overlaid and aligned prepreg platelets

Bifurcation analysis of density wave oscillations in natural circulation loop

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations – both, without temperature feedback and in the presence of temperature feedback – have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose (HR) neuronal model with the time-delayed feedback control. By stability and bifurcation analysis, we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation, and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current. For the purpose of control of chaos, we adopt the time-delayed feedback control, and convert chaos control to the Hopf bifurcation of the delayed feedback system. Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula. Based on this, we show the Hopf bifurcation curves in the two-parameter plane. Finally, some numerical simulations are carried out to support the theoretical results. It is shown that by appropriate choice of feedback gain and time delay, the chaotic orbit can be controlled to be stable. The adopted method in this paper is general and can be applied to other neuronal models. It may help us better understand the bifurcation mechanisms of neural behaviors.

A recent field manipulation on a terrestrial vertebrate showed that the fear of predators alone altered anti-predator defences to such an extent that it greatly reduced the reproduction of prey. Because fear can evidently affect the populations of terrestrial vertebrates, we proposed a predator-prey model incorporating the cost of fear into prey reproduction. Our mathematical analyses show that high levels of fear (or equivalently strong anti-predator responses) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, relatively low levels of fear can induce multiple limit cycles via subcritical Hopf bifurcations, leading to a bi-stability phenomenon. Compared to classic predator-prey models which ignore the cost of fear where Hopf bifurcations are typically supercritical, Hopf bifurcations in our model can be both supercritical and subcritical by choosing different sets of parameters. We conducted numerical simulations to explore the relationships between fear effects and other biologically related parameters (e.g. birth/death rate of adult prey), which further demonstrate the impact that fear can have in predator-prey interactions. For example, we found that under the conditions of a Hopf bifurcation, an increase in the level of fear may alter the direction of Hopf bifurcation from supercritical to subcritical when the birth rate of prey increases accordingly. Our simulations also show that the prey is less sensitive in perceiving predation risk with increasing birth rate of prey or increasing death rate of predators, but demonstrate that animals will mount stronger anti-predator defences as the attack rate of predators increases.

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis.

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results.

Coherence resonance (CR) describes a counterintuitive phenomenon in which the optimal oscillatory responses in nonlinear systems are shaped by a suitable noise amplitude. This phenomenon has been observed in neural systems. In this research, the generation of double coherence resonances (DCRs) due to white noise is investigated in a three-dimensional reduced Hodgkin-Huxley neuron model with multiple-timescale feature. We show that additive white noise can induce DCRs from the resting state near a subcritical Hopf bifurcation. The appearance of DCRs is related to the changes of the firing pattern aroused by the increases of the noise amplitude. The underlying dynamical mechanisms for the appearance of the DCRs and the changes of the firing pattern are interpreted using the phase space analysis and the dynamics of the stable focus-node near the subcritical Hopf bifurcation. We find that the multiple-timescale dynamics is essential for generating the DCRs and different firing patterns. The results not only present a case in which noise can induce DCRs near a Hopf bifurcation but also provide its dynamical mechanism, which enriches the phenomena in nonlinear dynamics and provides further understanding on the roles of noise in neural systems with multiple-timescale feature.

Gamma oscillation ([Formula: see text]) is an important neuronal rhythmic activity closely related to various brain functions, such as Parkinson's disease (PD). By identifying several direct glutamatergic projections from the cortical-thalamic system (CTS) to the subthalamic nucleus (STN)-external globus pallidus (GPe) circuit, we have established a novel cortical-basal ganglia-thalamus (CBGT) model. Within the CBGT model, we systematically investigate the dynamical mechanisms underlying the origin and control of [Formula: see text]. We find that significant narrowband [Formula: see text] in the 30-100 Hz range can emerge in the BG by adjusting the coupling weights and delays within the BG. Supercritical and subcritical Hopf bifurcations (SPHB and SBHB) can be used to explain the mechanisms underlying the origin of [Formula: see text]. We observe that all four direct glutamatergic projections from the CTS to the STN-GPe circuit can effectively inhibit [Formula: see text]. Interestingly, adjusting the activation levels of the thalamus and cortex can exert significant bidirectional Hopf bifurcation control over [Formula: see text] in the BG, through interaction among these four projections. This bidirectional Hopf bifurcation regulatory phenomenon exhibits good robustness with respect to parameters in the CBGT model, and parameters in the BG have a significant impact on the control patterns. Furthermore, we find that all pathways in the CTS actively participate in controlling [Formula: see text] in the BG, by adjusting the activation levels of cortical and thalamic nuclei. We observe the existence of significant high and low critical mean discharge rates (CMDR) in BG nuclei, as well as complex triggered mean discharge rates (TMDR) in cortical and thalamic nuclei, at the critical boundaries between [Formula: see text] and the stable state. These key dynamical indicators might provide testable foundations for experimental research. Notably, our computational findings suggest that cortical and thalamic nuclei could theoretically serve as potential targets for regulating oscillatory activity in the BG, providing a hypothesis for future experimental investigation into supplementary therapeutic approaches for PD.