Multiple limit cycles in a continuous culture vessel with variable yield

Zhu, Linsheng; Huang, Xuncheng

doi:10.1016/j.na.2005.05.049

Cited by 19 publications

(15 citation statements)

References 19 publications

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“…Since experiments have proved that the yield coefficient sometimes depends on the substrate concentration, for the convenience of discussion, most of researchers modelled the yield coefficient by a linear function [16,17]:…”

Section: Model Formulation and Preliminariesmentioning

confidence: 99%

“…If the yield coefficient increases linearly with the substrate concentration, then for certain parameter range, the stable rest state may undergo the Hopf bifurcation and a limit cycle appears [16,17,27]. …”

Section: Model Formulation and Preliminariesmentioning

confidence: 99%

“…Most of them assumed that the yield coefficient is constant. But some experiments have showed that the oscillatory behavior does exist in the chemostat [14][15][16][17]. Now the fact that the yield coefficient may depend on the substrate concentration has been well established by experiments [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

Guo

Chen

2009

J. Appl. Math. Comput.

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A turbidostat is an apparatus with feedback control system used to continuously culturing microorganisms. The dilution rate of the turbidostat can be regulated by the control system when the concentration of microorganism, detected by photoelectricity system or other devices, reaches a preset value. Based on the design ideas of the turbidostat, a differential equation with impulsive state feedback control is proposed for a kind of turbidostat system in this paper. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of periodic solution of order one are obtained according to the preset value and the types of the positive equilibrium of the corresponding system without impulsive control. Furthermore, it is pointed out that the system either tends to a stable state or has a periodic solution. Finally, the theoretical results are verified by numerical simulations.

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Section: Model Formulation and Preliminariesmentioning

confidence: 99%

Section: Model Formulation and Preliminariesmentioning

confidence: 99%

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Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

Guo

Chen

2009

J. Appl. Math. Comput.

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“…Chemostat models not only with constant yield but also with variable yield [19][20][21][22][23][24] are considered. Since actual experiments show that the constant yield cannot explain the oscillatory phenomenon in the chemostat, and the greater the nutrient concentrate is the lower the consuming rate is.…”

Section: Introductionmentioning

confidence: 99%

Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

Yao

Xiang

et al. 2018

Adv Differ Equ

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In this paper, dynamic behaviors of a turbidostat model with Tissiet functional response, linear variable yield and time delay are investigated. The existence and boundedness of solutions, the local asymptotic stability of its equilibria and the phenomenon of Hopf bifurcation for this system are considered. Using the Liapunov-LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Furthermore, based on some knowledge of limit set, we show the necessary and sufficient conditions of permanent of the turbidostat model. Finally, numerical simulations are offered to support our results.

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“…Although some efforts on multiple limit cycles in nonlinear systems are remarkable, such as the analysis on two limit cycles in the switching power converter [25] as well as the proof on existence conditions of three limit cycles in a continuous culture vessel system [26], many theoretical issues on the uniqueness and local stability of limit cycles in SM systems remain unanswered. Thus, theoretical studies on SM systems fall far behind their applications.…”

Section: Introductionmentioning

confidence: 99%

On multiple limit cycles in sliding-mode control systems via a generalized describing function approach

Shen

Qiu

2015

Nonlinear Dyn

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This paper proposes a generalized describing function (DF) approach to analyze a class of frequency-dependent nonlinearities which are hard to deal with by traditional DF approaches. The proposed approach can intuitively predict the number, stability, and parameters of limit cycles by graphical illustrations of amplitude-phase characteristics. When applied to a 2-relative-degree sampling output feedback system, multiple limit cycles are found in the sliding-mode control. Both behaviors of complex limit cycles and influences of initial conditions on limit cycle oscillations are analyzed. Results show that each limit cycle has its own attraction region which can be estimated by the chattering amplitudes of the state variables, and initial conditions in different attraction regions may induce multiple limit cycles in the same control system. The undesired limit cycles can be eliminated by parameter tuning, such as control gains and sampling periods, to obtain a globally stable one. Simulations and hardware experiments further confirm the validity of the generalized DF approach.

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Multiple limit cycles in a continuous culture vessel with variable yield

Cited by 19 publications

References 19 publications

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

On multiple limit cycles in sliding-mode control systems via a generalized describing function approach

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