BIFURCATION SOLUTIONS TO A BOUNDARY LAYER PROBLEM ARISING IN THE THEORY OFPOWER LAW FLUIDS

The present paper deals with a singular nonlinear boundary value problem arising in the theory of power law fluids, sufficient conditions for the existence of bifurcation solutions to the problem are obtained.

Stochastic Bifurcation of an SIS Epidemic Model with Treatment and Immigration

In this paper, we investigate an SIS model with treatment and immigration. Firstly, the two-dimensional model is simplified by using the stochastic averaging method. Then, we derive the local stability of the stochastic system by computing the Lyapunov exponent of the linearized system. Further, the global stability of the stochastic model is analyzed based on the singular boundary theory. Moreover, we prove that the model undergoes a Hopf bifurcation and a pitchfork bifurcation. Finally, several numerical examples are provided to illustrate the theoretical results. .

Stability and boundary equilibrium bifurcations of modified Chua's circuit with smooth degree of 3

Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.

Bifurcation Analysis for a Free Boundary Problem Modeling Growth of Solid Tumor with Inhibitors

This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.

DOUBLE BIFURCATION OF NONLINEAR DUFFING'S OSCILLATOR

The transition boundaries of period doubling on the physical parameter plane of a Duffing system are obtained by the general Newton′s method, and the motion on different areas divided by transition boundaries is studied in this paper. When the physical parameters transpass the boundaries, the solutions of period T =2π/ω will lose their stability, and the solutions of period T =2π/ω take place. Continuous period doubling bifurcations lead to chaos.

Bifurcation control of nonlinear oscillator in primary and secondary resonance

A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.

Numerical Study on the Bifurcation of the North Equatorial Current

A 1.5-layer reduced-gravity model forced by wind stress is used to study the bifurcations of the North Equatorial Current(NEC).The authors found that after removing the Ekman drift,the modelled circulations can serve well as a proxy of the SODA circulations on the σθ=25.0 kg m～-3 potential density surface based on available long-term reanalysis wind stress data.The modelled results show that the location of the western boundary bifurcation of the NEC depends on both zonal averaged and local zero wind stress curl latitude.The effects of the anomalous wind stress curl added in different areas are also investigated and it is found that they can change the strength of the Mindanao Eddy(ME),and then influence the interior pathway.

Bifurcation and catastrophe of seepage flow system in broken rock

The study of dynamical behavior of water or gas flows in broken rock is a basic research topic among a series of key projects about stability control of the surrounding rocks in mines and the prevention of some disasters such as water inrush or gas outburst and the protection of the groundwater resource. It is of great theoretical and engineering importance in respect of promo- tion of security in mine production and sustainable development of the coal industry. According to the non-Darcy property of seepage flow in broken rock dynamic equations of non-Darcy and non-steady flows in broken rock are established. By dimensionless transformation, the solution diagram of steady-states satisfying the given boundary conditions is obtained. By numerical analysis of low relaxation iteration, the dynamic responses corresponding to the different flow parameters have been obtained. The stability analysis of the steady-states indicate that a saddle-node bifurcaton exists in the seepage flow system of broken rock. Consequently, using catastrophe theory, the fold catastrophe model of seepage flow instability has been obtained. As a result, the bifurcation curves of the seepage flow systems with different control parameters are presented and the standard potential function is also given with respect to the generalized state variable for the fold catastrophe of a dynamic system of seepage flow in broken rock.

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit （oscillation） in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.

BIFURCATION ANALYSIS OF A DOUBLE PENDULUM WITH INTERNAL RESONANCE

By employing the normal form theory, the Hopf bifurcation and the transition boundary of an autonomous double pendulum with 1:1 internal resonance at the critical point is studied. The results are compared with numerical solutions. Further, by numerical methods, the road to chaos of a non-autonomous system is presented in the end.

Singularly perturbed bifurcation subsystem and its application in power systems

The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.

Characterization of static bifurcations for n-dimensional flows in the frequency domain

In this paper n-dimensional flows （described by continuous-time system） with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalism. Based on frequency domain approach, we prove some criterions for the saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and give an example to illustrate the efficiency of the result obtained.

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co_dimension two bifurcation system on a three_dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system_a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker_Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained.

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO_DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

For a real noise parametrically excited co_dimension two bifurcation system on a three_dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely,a zero_mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker_Planck operator.

Red blood cell partitioning and blood flux redistribution in microvascular bifurcation

This paper studies red blood cell （RBC） partitioning and blood flux redistribution in microvascular bifurcation by immersed boundary and lattice Boltzmann method. The effects of the initial position of RBC at low Reynolds number regime on the RBC deformation, RBC partitioning, blood flux redistribution and pressure distribution are discussed in detail. It is shown that the blood flux in the daughter branches and the initial position of RBC are important for RBC partitioning. RBC tends to enter the higher-flux-rate branch if the initial position of RBC is near the center of the mother vessel. The RBC may enter the lower-flux-rate branch if it is located near the wall of mother vessel on the lower-flux-rate branch side. Moreover, the blood flux is redistributed when an RBC presents in the daughter branch. Such redistribution is caused by the pressure distribution and reduces the superiority of RBC entering the same branch. The results obtained in the present work may provide a physical insight into the understanding of RBC partitioning and blood flux redistribution in microvascular bifurcation.

Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system

A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stochastic parametric excitations of equivalent conicity and suspension stiffness.The wheelset is systematized into a onedimensional(1D)diffusion process by using the stochastic average method,the behavior of the singular boundary is analyzed to determine the hunting stability condition of the wheelset system,and the critical speed of stochastic bifurcation is obtained.The stationary probability density and joint probability density are derived theoretically.Based on the topological structure change of the probability density function,the stochastic Hopf bifurcation form and bifurcation condition of the wheelset system are determined.The effects of stochastic factors on the hunting stability and bifurcation characteristics are analyzed,and the simulation results verify the correctness of the theoretical analysis.The results reveal that the boundary behavior of the diffusion process determines the hunting stability of the stochastic wheelset system,and the left boundary characteristic value cL=1 is the critical state of hunting stability.Besides,stochastic D-bifurcation and P-bifurcation will appear in the wheelset system,and the critical speeds of the two kinds of stochastic bifurcation decrease with the increase in the stochastic parametric excitation intensity.

Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.

Numerical Simulation of Flow Behavior in Basilar Bifurcation Aneurysms Based on 4-Dimensional Computed Tomography Angiography

Initiation, growth, and rupture of cerebral aneurysms are caused by hemodynamic factors. It is extensively accepted that the cerebral aneurysm wall is assumed to be rigid using computational fluid dynamics (CFD). Furthermore, fluid-structure interactions have been recently applied for simulation of an elastic cerebral aneurysm model. Herein, we examined cerebral aneurysm hemodynamics in a realistic moving boundary deformation model based on 4-dimensional computed tomographic angiography (4D-CTA) obtained by high time-resolution using numerical simulation. The aneurysm of the realistic moving deformation model based on 4D-CTA at each phase was constructed. The effect of small wall deformation on hemodynamic characteristics might be interested. So, four hemodynamic factors (wall shear stress, wall shear stress divergence, oscillatory shear index and residual residence time) were determined from the numerical simulation, and their behaviors were assessed in the basilar bifurcation aneurysm.

Periodical Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model with Feedback Control

The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.