Optimal residence time policy for product yield maximization in chemical reactors

This paper analyzes the problem of optimal reactor-type selection for the maximization of product yield in continuous, segregated flow, isothemal or adiabatic chemical reactors of given volume. The mathematical treatment is valid for any type of chemical reaction network with arbitrary kinetics. For fixed mean residence time, it is shown that maximum or minimum yield of a product can always be obtained exactly or within arbitrary approximation by a combination of two plug-flow reactors. The analysis assumes strictly segregated flow, but some of the conclusions reached do not depend on the extent of micromixing.This work was partly supported by the Hellenic Refineries of Aspropyrgos, Athens, through a fellowship to I. Andreou. The first author acknowledges helpful discussions with J. H. B. Kemperman, T. G. Hallam, and S. Papadopoulou.     

Stability and bifurcation of an unstructured model of a bioreactor with cell recycle

An unstructured model of a bioreactor with cell recycle and substrate inhibition kinetics is used to investigate the bifurcation and stability characteristics of this unit. The singularity theory used for this investigation allows a global analysis of steady-states multiplicity and the different bifurcation mechanisms occurring in the system including hysteresis and pitchfork. Analytical criteria are also derived for the safe operation of the reactor and to prevent wash-out conditions. The investigation of the dynamic bifurcation, on the other hand, shows that the model cannot exhibit periodic attractors with any growth kinetics model. The inability of this widely used model to exhibit periodicity despite the experimental results that support the existence of periodic behavior in many bioreactors suggests that new approaches are to be taken for the modeling.     

Spatio-temporal chaos in tubular chemical reactors with the recycle of mass

In this study the results are presented concerning the dynamics of homogeneous tubular chemical reactors with the recycle of mass. A detailed analysis shows that two types of dynamic bifurcation exist, namely, the flip bifurcation (FB) and the Hopf bifurcation (HB). It is demonstrated that each of these two types leads, for given values of the model parameters, to chaotic oscillations. Moreover, the Hopf bifurcation can also generate quasi-periodic solutions. The results are illustrated using temporal trajectories, bifurcation diagrams and Poincaré sections.     

Periodic and nonperiodic oscillatory behavior in a model for activated sludge reactors

A dynamic model for an activated sludge process is proposed to investigate the stability and bifurcation characteristics of this industrially important unit. The model is structured upon two processes: an intermediate participate product formation and active biomass synthesis processes. The growth kinetics expressions are based on substrate inhibition and noncompetitive inhibition of the intermediate product. The bifurcation analysis of the process model shows static as well as periodic behavior over a wide range of model parameters. The model also exhibits other interesting stability characteristics, including bistability and transition from periodic to nonperiodic behavior through period doubling and torus bifurcations. For some range of the reactor residence time the model exhibits chaotic behavior as well. Practical criteria are also derived for the effects of feed conditions and purge fraction on the dynamic characteristics of the bioreactor model.     

On stationary solutions of a nonlinear system of reactor dynamic equations with distributed parameters

We analyze a nonlinear stationary model of reactor dynamics with distributed parameters. We find sufficient conditions for the existence of bifurcation points in this system and study the behavior of solutions in a neighborhood of the bifurcation points. We prove the existence of countably many bifurcation points in the case of a homogeneous medium and obtain constructive estimates for the distance between the bifurcation points.     

Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition

In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator-prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T-H bifurcation (Turing-Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T-H bifurcation point, the normal form of the T-H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.     

Bifurcation and chaos in an epidemic model with nonlinear incidence rates

This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.     

Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy

In this paper, by using theories and methods of ecology and ODE, a two-prey one-predator system with Watt-type functional response and impulsive perturbations on the predator is established. The system is affected by impulse which can be considered as a control. Conditions for the permanence of the system are obtained. The numerical analysis is carried out to study the effects of perturbation varying parameters of the system. The system shows the rich dynamic behavior including quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises, etc.     

On the role of expectations in a dynamic Keynesian macroeconomic model

A simple Keynesian macroeconomic disequilibrium model with rationing is considered. This paper investigates the influence of different expectations hypotheses (regarding next period's goods prices) on the model's dynamic behavior when intertemporal substitution effects exist. While the bifurcation behavior is not qualitatively influenced when expectations are formed according to adaptive expectations or an unweighted-averages hypothesis, the dynamic behavior of the model can be changed in an essential way when pattern-recognition expectations are assumed. So-called perfect cyclic expectations can turn a previously chaotically moving economy into a system with a low-periodic motion in particular cases.     

The 3-D Hopf bifurcation in bio-reactor when one competitor produces a toxin

In this paper, a model of competition in the bio-reactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent is investigated. The conditions of the three dimensional Hopf bifurcation are obtained. The Hopf bifurcation implies the existence of limit cycles in the model that corresponds to the nonlinear oscillation in the reactor.     

Modelling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring

Using Hamilton variation principle, a nonlinear dynamic model of the system with a finite deforming Rayleigh beam clamped radially to the interior of a rotating rigid ring, under the assumption that the constitutive relation of the beam is linearly elastic, is discussed. The bifurcation behavior of the simple system with the Euler-Bernoulli beam is also discussed. It is revealed that these two models have no influence on the critical bifurcation value and buckling solution in the steady state. Then we use the assumption model method to analyse the bifurcation behavior of the steadily rotating Euler-Bernoulli beam and get two different types of bifurcation behavior which physically exist. Finite element method and shooting method are used to verify the analytical results. The numerical results confirm our research conclusion. Project supported by the National Natural Science Foundation of China (Grant No. 19332022) and Space High Technology Foundation of China.     

Mathematics analysis and chaos in an ecological model with an impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model with an impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest-level prey and mid-level predator eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

具有固定时滞的经济周期模型的动态分析

在经济活动中,投资行为和资本存量存在一定的时滞效应,这会影响经济周期模型的动态行为,进而使得投资政策对经济的稳定调整复杂化.考虑到资本存量的预期时间以及投资时滞对经济活动的影响,采用Hopf分岔理论,研究具有固定时滞的经济周期模型的均衡点的稳定性以及形成经济周期的条件.研究发现,投资过程中的投资时滞,以及投资决策中对于资本存量的预测时间构成经济周期产生的诱因;同时可通过政府投资政策调整达到预期均衡目标,这对保持经济周期稳定及经济政策制定有一定的指导作用.     

太湖大银鱼的种群数量模型

为研究大银鱼的种群数量,首先把大银鱼一年的生命周期分为三个阶段,在此基础上建立了一种既能在离散时间点上描述成年大银鱼的数量变化,又能描述每个繁殖期内从卵、幼鱼到成鱼演变过程的数学模型.最后,通过计算还发现离散时间的动力系统出现分岔现象,这就找到了大银鱼产量激烈变动的根本原因.     

周期激励下的前包钦格呼吸神经元的簇振荡现象研究

研究周期激励作用下的非自治前包钦格呼吸神经元模型,结果表明当外界激励频率与系统固有频率存在着量级差距时,系统可以产生典型的簇放电模式.由于激励频率远小于系统的固有频率,因此将整个周期激励项视为慢变参数,从而可以利用稳定性分析理论研究慢变参数变化下的平衡点的分岔类型,进一步应用快慢动力学分析方法给出簇模式产生的动力学机理.本文的结果说明外界激励对神经元的动力学行为有着重要影响,为进一步揭示呼吸节律的产生机制提供了重要帮助.     

Three-species food web model with impulsive control strategy and chaos

A three-species ecological model with impulsive control strategy is developed using the theory and methods of ecology and ordinary differential equation. Conditions for extinction of the system are given based on the theory of impulsive equation and small amplitude perturbation. Using comparison involving multiple Lyapunov functions, the system is shown to be permanent. Further, the influence of the impulsive perturbation on the inherent oscillation are studied numerically and is found to depict rich dynamics, such as the period-doubling bifurcation, the period-halving bifurcation, a chaotic band, a narrow or wide periodic window, and chaotic crises. In addition, the largest Lyapunov exponent is computed. This computation demonstrates the chaotic dynamic behavior of the model. The qualitative nature of concerned strange attractors is also investigated through their computed Fourier spectra. The foregoing results have the potential to be useful for the study of the dynamic complexity of ecosystems.     

大范围运动刚体上矩形薄板力学行为分析

采用Hamilton变分原理建立了大范围运动平板的动力学模型．从理论上证明了不同大范围运动状态下平板中既可存在动力刚化效应,也可存在动力软化效应,且动力软化效应还可使板的平衡状态发生分岔而失稳．采用假设模态法验证了理论分析结果并得到了分岔临界值和近似后屈曲解．     

On the oil-whipping of a rotor-bearing system by a continuum model

The nonlinear dynamic behavior of a rotor-bearing system is analyzed based on a continuum model. The finite element method is adopted in the analysis. Emphasis is placed on the so-called “oil-whip phenomena” which might lead to the failure of the rotor system. The dynamic response of the system in unbalanced conditions is approached by a direct integration method. It is found that a typical “oil-whip phenomenon” is successfully simulated, and the effect of the refinement of the finite element mesh is also checked. Furthermore, the bifurcation behavior of the oil-whip phenomenon that is of much concern in recent nonlinear dynamics research is analyzed. The rotor-bearing system is also examined by a simple discrete model. Significant differences are found between these two models. It is suggested that a careful examination should be made in modeling the nonlinear dynamic behavior of a rotor system.