Solution and transcritical bifurcation of Burgers equation

Burgers equation is reduced into a first-order ordinary differential equation by using travelling wave transformation and it has typical bifurcation characteristics. We can obtain many exact solutions of the Burgers equation, discuss its transcritical bifurcation and control dynamical behaviours by extending the stable region. The transcritical bifurcation exists in the （2 ＋ 1）-dimensional Burgers equation.

Dynamic Analysis of an Algae-Bacteria Ecological Model

In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.

Comparative analysis of CO_2-based transcritical Rankine cycle and HFC245fa-based subcritical organic Rankine cycle using low-temperature geothermal source

A detailed thermodynamic and techno-economic comparison is presented for a CO2-based transcritical Rankine cycle and a subcritical organic Rankine cycle (ORC) using HFC245fa (1,1,1,3,3-pentafluoro-propane) as the working fluid driven by the low-temperature geothermal source,in order to determine the configuration that presents the maximum net power output with a minimum investment.The evaluations of both Rankine cycles have been performed based on equal thermodynamic mean heat rejection temperature by varying certain system operating parameters to achieve each Rankine cycle's optimum design at various geothermal source temperature levels ranging from 80oC to 120oC.The results obtained show that the optimum ther-modynamic mean heat injection temperatures of both Rankine cycles are distributed in the scope of 55% to 65% of a given geothermal source temperature level,and that the CO2-based transcritical Rankine cycle presents 3% to 7% higher net power output,84% reduction of turbine inlet volume flow rate,47% reduction of expansion ratio and 1.68 times higher total heat transfer capacity compared with the HFC245fa-based subcritical ORC.It is also indicated that using the CO2-based transcritical system can reduce the dimension of turbine design.However,it requires larger heat transfer areas with higher strength heat exchanger materials because of the higher system pressure.

Experimental evaluation of the effect of an internal heat exchanger on a transcritical CO_2 ejector system

This study presents experimental results focused on a performance comparison of a transcritical CO2 ejector system without an internal heat exchanger(IHX) (EJE-S) to a transcritical CO2 ejector system with an IHX(EJE-IHX-S) . The comparison includes the effects of changes in operating conditions such as cooling water flow rate and inlet temperature. Experiments are conducted to assess the influence of the IHX on the heating coefficient of performance(COPr) ,heating capacity,entrainment ratio,pressure lift,and other parameters. The primary flow rate of the EJE-IHX-S is higher than that of the EJE-S. The pressure lift and actual ejector work recovery are reduced when the IHX is added to the transcritical CO2 ejector system. Using a more practical performance calculation,the compression ratio in the EJE-S is reduced by 10.0%-12.1%,while that of EJE-IHX-S is reduced only by 5.6%-6.7% compared to that of a conventional transcritical CO2 system. Experimental results are used to validate the findings that the IHX weakens the contribution of the ejector to the system performance.

Numerical investigation of transcritical liquid film cooling in a methane/oxygen rocket engine

Transcritical film cooling was investigated by numerical study in a methane cooled methane/oxygen rocket engine.The respective time-averaged Navier-Stokes equations have been solved for the compressible steady three-dimensional(3-D) flow.The flow field computations were performed using the semi-implicit method for pressure linked equation(SIMPLE) algorithm on several blocks of nonuniform collocated grid.The calculation was conducted over a pressure range of 202 650.0 Pa to 1.2×107 Pa and a temperature range of 120.0 K to 3 568.0 K.Twenty-nine different cases were simulated to calculate the impact of different factors.The results show that mass flow rate,length,diameter,number and diffused or convergence of film jet channel,injection angle and jet array arrangements have great impact on transcritical film cooling effectiveness.Furthermore,shape of the jet holes and jet and crossflow turbulence also affect the wall temperature distribution.Two rows of film arranged in different axial angles and staggered arrangement were proposed as new liquid film arrangement.Different radial angles have impact on the film cooling effectiveness in two row-jets cooled cases.The case of in-line and staggered arrangement are almost the same in the region before the second row of jets,but a staggered arrangement has a higher film cooling effectiveness from the second row of jets.

Carbon Dioxide as Working Fluids in Transcritical Rankine Cycle for Diesel Engine Multiple Waste Heat Recovery in Comparison to Hydrocarbons

In consideration of the high-temperature characteristic of engine's waste heat and stricter environmental regulations, natural substance, including CO_2 and hydrocarbons, have been treated as promising working fluid for diesel engine waste heat recovery due to its environment friendly and excellent physical and chemical properties. This paper presented a comprehensive performance analysis on transcritical Rankine cycles for diesel engine multiple waste heat recovery using hydrocarbons and CO_2 as working fluid. The optimal turbine inlet pressures corresponding to maximum net power output, maximum exergy efficiency and minimum electricity production cost(EPC) were obtained. The effect of working fluid on these optimal pressures has been discussed. For fluids with low critical temperature, the optimal pressure corresponding to maximum net power output is lower than the one for maximum exergy efficiency, while the opposite results can be found for fluid with high critical temperature. Then, the effect of various working fluid properties in transcritical cycle performance is discussed. Comparison results show that CO_2 obtains only more power output than Ethane, Propane and Propene, but CO_2 is capable of absorbing more energy from engine coolant and regeneration heat with comparable total heat transfer areas and has an advantage in turbine size, particularly for hydrocarbons with high critical temperature.

Extending Slow Manifold Near Generic Transcritical Canard Point

We consider the dynamics of planar fast-slow systems near generic transcritical type canard point. By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.

Thermodynamic Evaluation of Transcritical CO_(2) Heat Pump Considering Temperature Matching under the Constraint of Heat Transfer Pinch Point

The non-linear temperature glide in the supercritical CO_(2) cooling process makes the heat transfer pinch point of heat exchanger show multiplicity,like size,location distribution and quantity,which makes the thermodynamic performance of the CO_(2) transcritical cycle more complex and eventually affects the evaluation of the optimal operating state of the system.Based on the second law of thermodynamics and the constraints of heat transfer pinch,a thermodynamic evaluation method of CO_(2) transcritical heat pump system was proposed according to the degree of temperature matching.The influence mechanism of multi-characteristic change of heat transfer pinch point on temperature matching degree and the effect of temperature matching degree on thermodynamic performance of CO_(2) transcritical heat pump system were discussed.The relationship between temperature matching degree,COP and exergy efficiency of the system was analyzed.It is considered that the change of temperature matching index value can clearly characterize the change trends of COP and exergy efficiency.That is,the smaller the temperature matching degree is,the closer the temperature distribution of heat transfer fluids on both sides of the heat exchanger is to Lorenz cycle,and the greater the COP and exergy efficiency are.Furthermore,by monitoring the outlet temperature of the CO_(2) cooler,which has an essential relationship with the temperature matching degree during the heat exchange process,the deviation between actual and optimal working condition can be judged online,which is beneficial to real-time evaluation of the working state of the system.

一类离散不可逆平面映射的分支

本文主要研究的是一类离散的不可逆平面映射,当参数变化时,该映射动态的变化情况。首先,对不可逆平面映射的不动点的存在性及稳定性进行了分析,得到不可逆平面映射有两个不动点,通过对参数a,b进行分析,分别得出两个不动点稳定性的条件;其次,通过对参数b进行控制,从而导出flip分支、transcritical分支及Hopf分支存在的充分条件;最后通过数值模拟,验证了这些分支存在条件的理论结果。

混杂系统的几种分岔

总结了常用的混杂系统的数学模型,并针对微分差分代数形式的混杂系统给出Transcritical分岔,Pitchfork分岔和奇异诱导分岔分析.

具有非线性传染率的SIS网络传染病模型的稳定性和分支分析

研究了一类具有非线性传染率的SIS网络传染病模型的动力学行为,给出传播阈值λ_c=〈k〉/<k(k-1)φ(k)>.结果表明,当β_0<λ_c时,无病平衡点E_0=0局部稳定;当β_0>λ_c时,无病平衡点E_0=0不稳定;进一步分析,当β_0=λ_c时,系统在E_0=0处出现Transcritical分支.

The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time

In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map. Then we find out its equilibrium points and eigenvalues. From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point. Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem.

The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination

In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.

一类带恐惧效应的三物种食物链模型的分支分析

主要研究了恐惧效应对三物种食物链模型中分支动态的影响,运用中心流形定理和局部分支理论分析了Hopf分支、transcritical分支和saddle-node分支的存在性,并给出相应的数值模拟结果。研究结果表明,分支是种群发生失稳和周期性振荡的根本原因,从而揭示了恐惧效应是维持种群稳定的重要因素。

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.

The Dynamics and Bifurcation Control of a Singular Biological Economic Model

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.

Thermodynamic Performance Comparison and Optimization of sCO_(2)Brayton Cycle,tCO_(2)Brayton Cycle and tCO_(2)Rankine Cycle

In this paper,to further improve thermodynamic performance of supercritical carbon dioxide cycle,simple/recompression transcritical carbon dioxide Brayton cycle(STBC/RTBC)and simple/recompression transcritical carbon dioxide Rankine cycle(STRC/RTRC)are proposed.Thermal and exergy performance analysis and optimization for the above four transcritical CO_(2)cycles and simple/recompression supercritical cycle(SSBC/RSBC)are conducted.The effect of key thermodynamic parameters on those CO_(2)cycle performance is studied.Results indicate that the improvements of thermodynamic performance of CO_(2)cycle are obvious when transcritical Brayton and Rankine cycle are applied in it.Within the same range of optimization variables,the maximum thermal efficiency improvements of RTRC and RTBC are 4.98%and 3.6%,and maximum exergy efficiency improvements of RTRC and RTBC are 7.08%and 5.13%when compared with RSBC.Moreover,the thermodynamic performances of STBC and STRC are also outstanding than that of SSBC.This work provides a way to further improve the thermodynamic performance of CO_(2)power cycle.

Bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.

Analytical bifurcation and strong resonances of a discrete Bazykin-Berezovskaya predator-prey model with Allee effect

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin-Berezovskaya predator-prey model in depth using analytical and numerical bifurcation analysis.The stability conditions of fixed points,codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied.This model exhibits transcritical,fip,Neimark-Sacker,and 1:2,1:3,1:4 strong resonances.The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory.For each bifurcation,various types of critical states are calculated,such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point.To validate our analytical findings,the bifurcation curves of fixed points are determined by using MatcontM.

Dynamical study of a predator-prey system with Michaelis-Menten type predator-harvesting

The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the predation process,which benefits the predators by developing fear upon their prey.This study investigates the dynamical behavior of a modified LV-type predator-prey system with Michaelis-Menten-type harvesting of predators where predators adopt cooperation strategy during hunting.The ecologically feasible steady states of the system and their asymptotic stabilities are explored.The local codimension one bifurcations,viz.transcritical,saddle-node and Hopf bifurcations,that emerge in the system are investigated.Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation.A backward Hopfbifurcation is detected when the harvesting effort is increased,which destabilizes the system by generating periodic solutions.The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient.Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations.Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased.We have also explored codimension two bifurcations viz.the generalized Hopf and the Bogdanov-Takens bifurcation exhibit by the system.To visualize the dynamical behavior of the system,numerical simulations are conducted using an ecologically plausible parameter set.The existence of the bionomic equilibrium of the model is analyzed.Moreover,an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.