Criticality Of Hopf Bifurcation Research Articles

Safety estimation for a new model of regenerative and frictional cutting dynamics

In our previous study on regenerative and frictional cutting dynamics, a new model considering both time-delayed regenerative effect and Stribeck effect as sources of cutting instability has been proposed [7], which has improved the prediction of linear stability in the zone of low cutting velocity. Based on the new model, this investigation explores complex nonlinear cutting dynamics. More specifically, the criticality of Hopf bifurcation on the cutting stability boundaries is studied by perturbation analysis showing the co-existence of cutting stationary and chatter in the linearly stable region, i.e., the unsafe zones (UZs). Then this analytical estimation is compared with numerical simulation revealing a possibility of underestimation due to the large-amplitude frictional chatter entering the stable region, which expands the UZs. Beside this local perturbation analysis, global bifurcation diagrams are constructed by numerical simulation, yielding various complex responses including multiple stability, regenerative chatter with loss of tool-workpiece contact and stick-slip frictional vibration. Finally, the cutting safety in the UZs is studied based on basin stability estimation, where the initial conditions are approximated by Fourier series and chatter occurrence is estimated via Monte Carlo simulation. It is found that from the statistical point of view, the large-amplitude frictional chatter hardly influences the UZs.

On the first Liapunov coefficient formula of 3D Lotka-Volterra equations with applications to multiplicity of limit cycles

This paper provides the first Liapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincaré-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman's classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Liapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all 2×2 principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty.

Change in criticality of Hopf bifurcation in a time-delayed cancer model

Change in criticality of Hopf bifurcation in a time-delayed cancer model

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen’s manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.

Change of criticality in a prototypical thermoacoustic system.

In this paper, we report on the existence of the phenomenon of change of criticality in a horizontal Rijke tube, a prototypical thermoacoustic system. In the experiments, the phenomenon is shown to occur as the criticality of the Hopf bifurcation changes with varying air flow rates in the system. The dynamics of a nonlinear system exhibiting Hopf bifurcation can be described using a Stuart-Landau equation (SLE) in the vicinity of the bifurcation point. The criticality of Hopf bifurcations can be determined by the Landau constant of the Stuart-Landau equation, which represents the effect of nonlinearities in the system. We propose an SLE to model the bifurcations seen in the horizontal Rijke tube. We identify a rescaled version of Strouhal number as the Landau constant, which determines the criticality of the bifurcation in the present study.

Stability and bifurcation analyses of chatter vibrations in a nonlinear cylindrical traverse grinding process

In this study, stability and bifurcation analyses are performed on a cylindrical traverse grinding process in order to investigate its nonlinear chatter behaviors. The grinding model system under consideration appears to be a set of autonomous doubly regenerative delay differential equations. The linear stability boundaries of this grinding system are first evaluated by performing an eigen-analysis on the linearized system. In this stability analysis, a boundness condition for the chatter frequency is obtained and is used to avoid difficulties in identifying the stability boundary caused by the infinite-dimensional nature of the delayed system. The resulting linear stability diagrams are illustrated in the parametric windows of interest. Based on these stability diagrams, bifurcation analyses are conducted by the methods of multiple scales and harmonic balance (MMS and MHB) in order to investigate the local and global chatter behaviors. The criticality of Hopf bifurcation is analytically determined based on the normal form equations of the grinding system through the MMS. Furthermore, the codimension-two bifurcations of equilibrium, such as the Bautin and Hopf–Hopf bifurcations, are also identified in this study. For large-amplitude chatter behaviors, the periodic solutions bifurcated from the critical equilibrium are calculated by the MHB. The cyclic fold bifurcation of the limit cycle is newly identified. In these bifurcation analyses, the nonlinear chatter behaviors of the present grinding system are examined and discussed through the resulting bifurcation diagrams of limit cycles, the Bautin bifurcation diagram, and the corresponding phase portraits. These results are validated by comparison with those obtained through direct numerical integration.

Stability and bifurcation of a two-neuron network with distributed time delays

Abstract In this paper we study the stability and bifurcation of the trivial solution of a two-neuron network model with distributed time delays. This model consists of two identical neurons, each possessing nonlinear instantaneous self-feedback and connected to the other neuron with continuously distributed time delays. We first examine the local asymptotic stability of the trivial solution by studying the roots of the corresponding characteristic equation, and then describe the stability and instability regions in the parameter space consisting of the self-feedback strength and the product of the connection strengths between the neurons. It is further shown that the trivial solution may lose its stability via a certain type of bifurcation such as a Hopf bifurcation or a pitchfork bifurcation. In addition, the criticality of Hopf bifurcation is investigated by means of the normal form theory. We also provide numerical evidence to support our theoretical analyses.