scholarly journals Global Stability Analysis of a Bioreactor Model for Phenol and Cresol Mixture Degradation

Processes ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 124
Author(s):  
Neli Dimitrova ◽  
Plamena Zlateva
Keyword(s):  
Specific Growth ◽  
Equilibrium Points ◽  
Nonlinear Ordinary Differential Equations ◽  
The Novel ◽  
Model Design ◽  
Stirred Bioreactor ◽  
Global Stability Analysis ◽  
Local Asymptotic Stability ◽  
Model Dynamics ◽  
Stability And Bifurcations

We propose a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The model is described by three nonlinear ordinary differential equations. The novel idea in the model design is the biomass specific growth rate, known as sum kinetics with interaction parameters (SKIP) and involving inhibition effects. We determine the equilibrium points of the model and study their local asymptotic stability and bifurcations with respect to a practically important parameter. Existence and uniqueness of positive solutions are proved. Global stabilizability of the model dynamics towards equilibrium points is established. The dynamic behavior of the solutions is demonstrated on some numerical examples.

Dynamic Analysis of the Melanoma Model: From Cancer Persistence to Its Eradication

2017 ◽  
Vol 27 (10) ◽  
pp. 1750151 ◽  
Author(s):  
Konstantin E. Starkov ◽  
Laura Jimenez Beristain
Keyword(s):  
Equilibrium Point ◽  
Global Attractivity ◽  
Cytotoxic T Cells ◽  
Equilibrium Points ◽  
Upper And Lower Bounds ◽  
Cellular Immunotherapy ◽  
Global Dynamics ◽  
Global Stability Analysis ◽  
Local Asymptotic Stability ◽  
Melanoma Model

In this paper, we study the global dynamics of the five-dimensional melanoma model developed by Kronik et al. This model describes interactions of tumor cells with cytotoxic T cells and respective cytokines under cellular immunotherapy. We get the ultimate upper and lower bounds for variables of this model, provide formulas for equilibrium points and present local asymptotic stability/hyperbolic instability conditions. Next, we prove the existence of the attracting set. Based on these results we come to global asymptotic melanoma eradication conditions via global stability analysis. Finally, we provide bounds for a locus of the melanoma persistence equilibrium point, study the case of melanoma persistence and describe conditions under which we observe global attractivity to the unique melanoma persistence equilibrium point.

scholarly journals Time-Delayed Bioreactor Model of Phenol and Cresol Mixture Degradation with Interaction Kinetics

Water ◽  
2021 ◽  
Vol 13 (22) ◽  
pp. 3266
Author(s):  
Milen Borisov ◽  
Neli Dimitrova ◽  
Plamena Zlateva
Keyword(s):  
Growth Rate ◽  
Time Delay ◽  
Mutual Influence ◽  
Equilibrium Points ◽  
Real Life ◽  
Biomass Growth ◽  
Stirred Bioreactor ◽  
Biological Time ◽  
Local Asymptotic Stability ◽  
Discrete Time Delay

This paper is devoted to a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The biomass specific growth rate is presented as sum kinetics with interaction parameters (SKIP). A discrete time delay is introduced and incorporated into the biomass growth response. These two aspects—the mutual influence of the two substrates and the natural biological time delay in the biomass growth rate—are new in the scientific literature concerning bioreactor (chemostat) models. The equilibrium points of the model are determined and their local asymptotic stability as well as the occurrence of local Hopf bifurcations are studied in dependence on the delay parameter. The existence and uniqueness of positive solutions are established, and the global stabilizability of the model dynamics is proved for certain values of the delay. Numerical simulations illustrate the global behavior of the model solutions as well as the transient oscillations as a result of the Hopf bifurcation. The performed theoretical analysis and computer simulations can be successfully used to better understand the biodegradation dynamics of the chemical compounds in the bioreactor and to predict and control the system behavior in real life conditions.

scholarly journals Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Kaibin Chu ◽  
Zhengwei Zhu ◽  
Hui Qian ◽  
Huagan Wu
Keyword(s):  
Control Function ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Stable Node ◽  
The Novel ◽  
Analog Multiplier ◽  
Piecewise Linearity ◽  
Index 2 ◽  
Hardware Circuit ◽  
Zero Index

With new three-segment piecewise-linearity in the classic Chua’s system, two new types of 2-scroll and 3-scroll Chua’s attractors are found in this paper. By changing the outer segment slope of the three-segment piecewise-linearity as positive, the new 2-scroll Chua’s attractor has emerged from one zero index-1 saddle-focus and two symmetric stable nonzero node-foci. In particular, by newly introducing a piecewise-linear control function, an improved Chua’s system only with one zero index-2 saddle-focus and two stable nonzero node-foci is constructed, from which a 3-scroll Chua’s attractor is converged. Some remarks for Chua’s nonlinearities and the generating chaotic attractors are discussed, and the stabilities at the three equilibrium points are then analyzed, upon which the emerging mechanisms of the novel 2-scroll and 3-scroll Chua’s attractors are explored in depth. Furthermore, an analog electronic circuit built with operational amplifier and analog multiplier is designed and hardware circuit experiments are measured to verify the numerical simulations. These novel 2-scroll and 3-scroll Chua’s attractors reported in this paper are completely different from the classic Chua’s attractors, which will enrich the dynamics of the classic Chua’s system.

QUENCHING LORENZIAN CHAOS

2004 ◽  
Vol 14 (08) ◽  
pp. 2875-2884 ◽  
Author(s):  
RAYMOND HIDE ◽  
PATRICK E. McSHARRY ◽  
CHRISTOPHER C. FINLAY ◽  
GUY D. PESKETT
Keyword(s):  
Parameter Space ◽  
Control Parameter ◽  
Nonlinear Dynamical Systems ◽  
General Interest ◽  
Nonlinear Ordinary Differential Equations ◽  
The Novel ◽  
Nonlinear Dynamical ◽  
F Region ◽  
Dimensionless Variables ◽  
Autonomous Set

How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on "nonlinear quenching" [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) [Formula: see text], ẏ=-xzq+bx-y and ż=xyq-cz (where (x, y, z) are time(t)-dependent dimensionless variables and [Formula: see text], etc.) in representative cases when q, the "quenching function", satisfies q=1-e+ey with 0≤e≤1. Control parameter space based on a,b and c can be divided into two "regions", an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The "Hopf boundary" between the two regions is located where b=bH(a, c; e) (say), with the much studied point (a, b, c)=(10, 28, 8/3), where the persistent "Lorenzian" chaos that arises in the case when e=0 was first found lying close to b=bH(a, c; 0). As e increases from zero the S-region expands in total "volume" at the expense of F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the entire parameter space.

Non-linear dynamic behaviour of rotor—bearing system trained by bevel gears

2008 ◽  
Vol 222 (4) ◽  
pp. 617-627 ◽  
Author(s):  
M Li
Keyword(s):  
Equilibrium Points ◽  
Rotor System ◽  
Design Stage ◽  
Oil Film ◽  
Bevel Gears ◽  
Linear Dynamic ◽  
Non Linear ◽  
Rotor Bearing ◽  
Stability And Bifurcations ◽  
Bearing System

The vibrations of parallel geared rotor—bearing system have been intensively discussed; however, little attention has been paid to the dynamic analysis of angled bevel-geared system supported on journals. In the present work, the non-linear dynamics of a bevel-geared rotor system on oil film bearings is studied. First, the dynamic model is developed under some assumptions, such as rigid rotors, short-bearings, small teeth errors, and so forth. Then, the non-linear dynamic behaviours of both the balanced and unbalanced rotor system are analysed, respectively, in which the equilibrium points, limit cycles, their stability, and bifurcations are paid more attention. Numerical results show that in the bevel-geared rotor system under the action of non-linear oil film forces there exists a series of complex non-linear dynamic phenomena of rotor orbits, such as Hopf bifurcation, torus-doubling bifurcation, and jump phenomenon. All these features can help us to understand the dynamic characteristics of bevel-geared rotor—bearing system at design stage and during running period. Finally, some concerned problems during the investigation are also present.

scholarly journals On Stability Analysis of Higher-Order Rational Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Abdul Khaliq ◽  
H. S. Alayachi ◽  
M. S. M. Noorani ◽  
A. Q. Khan
Keyword(s):  
Stability Analysis ◽  
Equilibrium Points ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Local Asymptotic Stability ◽  
Recursive Sequence ◽  
Rational Difference Equation ◽  
Theoretical Results

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞, α, β, γ, δ∈0,∞, and r>p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1, and w0 are positive real numbers. Some numerical examples are given to verify our theoretical results.

Four-Wing Hidden Attractors with One Stable Equilibrium Point

2020 ◽  
Vol 30 (06) ◽  
pp. 2050086 ◽  
Author(s):  
Quanli Deng ◽  
Chunhua Wang ◽  
Linmao Yang
Keyword(s):  
Equilibrium Point ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Stable Node ◽  
Stable Equilibrium Point ◽  
The Novel ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Attraction Basin

Although multiwing hidden attractor chaotic systems have attracted a lot of interest, the currently reported multiwing hidden attractor chaotic systems are either with no equilibrium point or with an infinite number of equilibrium points. The multiwing hidden attractor chaotic systems with stable equilibrium points have not been reported. This paper reports a four-wing hidden attractor chaotic system, which has only one stable node-focus equilibrium point. The novel system can also generate a hidden attractor with one-wing and hidden attractors with quasi-periodic and periodic coexistence. In addition, a self-excited attractor with one-wing can be generated by adjusting the parameters of the novel system. The hidden attractors of the novel system are verified by the cross-section of attraction basins. And the hidden behavior is investigated by choosing different initial states. Moreover, the coexisting transient four-wing phenomenon of the self-excited one-wing attractor system is studied by the time domain waveforms and attraction basin. The dynamical characteristics of the novel system are studied by Lyapunov exponents spectrum, bifurcation diagram and Poincaré map. Furthermore, the novel hidden attractor system with four-wing and one-wing are implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

Partial state feedback sliding mode control for double-pendulum overhead cranes with unknown disturbances

2022 ◽  
pp. 095440622110520
Author(s):  
Qingrong Chen ◽  
Wenming Cheng ◽  
Jiahui Liu ◽  
Run Du
Keyword(s):  
Nonlinear Dynamics ◽  
State Feedback ◽  
Sliding Mode ◽  
Equilibrium Points ◽  
Sliding Mode Controller ◽  
Double Pendulum ◽  
The Novel ◽  
Overhead Cranes ◽  
Unknown Disturbances ◽  
Partial State

In this paper, a novel sliding mode controller which requires partial state feedback is proposed for double-pendulum overhead cranes subject to unknown payload parameters and unknown external disturbances. Firstly, it is theoretically proved that the hook and payload tend to their respective equilibrium points concurrently. Secondly, a decoupling transformation is performed on the original nonlinear dynamics of double-pendulum overhead cranes. The novel sliding mode controller that does not require the prior information and motion signals of the payload is designed based on the decoupled nonlinear dynamics. Then, the asymptotic stability of the equilibrium point of double-pendulum overhead cranes is proved by rigorous analysis. Finally, several simulations are conducted to validate the effectiveness and robustness of the proposed controller.

Conformable Fractional SEIR Epidemic Model With Treatment

2021 ◽  
Vol 14 ◽  
Author(s):  
Arti Malik ◽  
Nitendra Kumar ◽  
Khursheed Alam
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Asymptotic Stability ◽  
Epidemic Model ◽  
Analytical Study ◽  
Equilibrium Points ◽  
Treatment Modalities ◽  
Local Asymptotic Stability ◽  
Fractional Differential ◽  
Disease Free

Background: The present paper is based on models of conformable fractional differential equation to describe the dynamics of certain epidemics. Methods: In this paper we have divided the population in the susceptible, exposed, infectious, recovered and also describe the treatment modalities. Results: The analytical study of the model show two equilibrium points (disease free equilibrium and endemic equilibrium). Conclusion: For both cases local asymptotic stability has been proven. In the conclusion we have presented the numerical simulation.

DYNAMICAL ANALYSIS OF A ALLELOPATHIC PHYTOPLANKTON MODEL

2006 ◽  
Vol 14 (02) ◽  
pp. 205-217 ◽  
Author(s):  
MALAY BANDYOPADHYAY
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Equilibrium Points ◽  
Toxic Substance ◽  
Model System ◽  
Dynamical Analysis ◽  
Local Asymptotic Stability ◽  
Stabilizing Effect

In this paper we have considered a two-species competitive phytoplankton system with one toxin producing phytoplankton. Local asymptotic stability of various equilibrium points are considered to understand the effect of toxic substance on the dynamics of the model system. By using a suitable Lyapunov function we have observed that the toxic substance has some stabilizing effect on the dynamics of model system.

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