Sufficient conditions for uniqueness and the local asymptotic stability of adiabatic tubular reactors

AIChE Journal ◽  
1972 ◽  
Vol 18 (5) ◽  
pp. 1079-1081 ◽  
Author(s):  
C. T. Liou ◽  
H. C. Lim ◽  
W. A. Weigand
Keyword(s):  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Local Asymptotic Stability ◽  
Tubular Reactors

Discontinuous Lyapunov functions for a class of piecewise affine systems

2018 ◽  
Vol 41 (3) ◽  
pp. 729-736 ◽  
Author(s):  
Farideh Cheraghi-Shami ◽  
Ali-Akbar Gharaveisi ◽  
Malihe M Farsangi ◽  
Mohsen Mohammadian
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Monotonicity Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Local Asymptotic Stability

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

Dynamic Analysis of a Delayed Fractional-Order SIR Model with Saturated Incidence and Treatment Functions

2018 ◽  
Vol 28 (14) ◽  
pp. 1850180 ◽  
Author(s):  
Xinhe Wang ◽  
Zhen Wang ◽  
Xia Huang ◽  
Yuxia Li
Keyword(s):  
Asymptotic Stability ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Local Asymptotic Stability ◽  
Proposed Model ◽  
Saturated Incidence ◽  
Theoretical Results

In this paper, a delayed fractional-order SIR (susceptible, infected, and removed) epidemic model with saturated incidence and treatment functions is presented. Firstly, the non-negativity and boundedness of solutions of the proposed model are proved. Next, some sufficient conditions are established to ensure the local asymptotic stability of the disease-free equilibrium point [Formula: see text] and the endemic equilibrium point [Formula: see text] for any delay. Meanwhile, global asymptotic stability of the endemic equilibrium point [Formula: see text] is investigated by constructing a suitable Lyapunov function. Some sufficient conditions are established for the global asymptotic stability of this endemic equilibrium point. Finally, some numerical simulations are illustrated to verify the correctness of the theoretical results.

scholarly journals Dynamic Behaviors of a Leslie-Gower Ecoepidemiological Model

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Aihua Kang ◽  
Yakui Xue ◽  
Jianping Fu
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Lyapunov Functions ◽  
Total Population ◽  
Sufficient Conditions ◽  
Prey Population ◽  
Dynamic Behaviors ◽  
Interior Equilibrium ◽  
Local Asymptotic Stability

A Leslie-Gower ecoepidemic model with disease in the predators is constructed and analyzed. The total population is subdivided into three subclasses, namely, susceptible predator, infected predator, and prey population. The positivity, boundness of solutions, and the existence of the equilibria are studied, and the sufficient conditions of local asymptotic stability of the equilibria are obtained by the Routh-Hurwitz criterion. We analyze the global stability of the interior equilibria by using Lyapunov functions. It is observed that a Hopf bifurcation may occur around the interior equilibrium. At last, numeric simulations are performed in support of the feasibility of the main result.

Simple algebraic necessary and sufficient conditions for Lyapunov stability of a Chen system and their applications

2017 ◽  
Vol 40 (7) ◽  
pp. 2200-2210 ◽  
Author(s):  
Guopeng Zhou ◽  
Xiaoxin Liao ◽  
Bingji Xu ◽  
Pei Yu ◽  
Guanrong Chen
Keyword(s):  
Asymptotic Stability ◽  
Exponential Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Feedback ◽  
Chen System ◽  
Local Asymptotic Stability ◽  
Positive Elements ◽  
Necessary And Sufficient

In this paper, we study the Lyapunov stability problem of a Chen chaotic system. Because of the positive elements of the main diagonal of a linearized Chen system, compared to the coefficient of a linearized Lorenz system which are all negative, it is more difficult to deal with the stability analysis. Since it has the properties of invariance and symmetry, different Lyapunov functions in different regions are constructed to solve stability problems with geometric and algebraic methods. Then, simple algebraic necessary and sufficient conditions of global exponential stability, global asymptotic stability and global instability of equilibrium [Formula: see text] are proposed. We obtain the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability and local instability of equilibria [Formula: see text]. Furthermore, the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria.

scholarly journals Some Qualitative Behavior of Solutions of General Class of Difference Equations

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 585 ◽  
Author(s):  
Osama Moaaz ◽  
Dimplekumar Chalishajar ◽  
Omar Bazighifan
Keyword(s):  
Periodic Solution ◽  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equations ◽  
General Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Qualitative Behavior ◽  
Local Asymptotic Stability ◽  
Necessary And Sufficient

In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones.

scholarly journals On neutral-delay two-species Lotka-Volterra competitive systems

1991 ◽  
Vol 32 (3) ◽  
pp. 311-326 ◽  
Author(s):  
Y. Kuang
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Qualitative Behavior ◽  
Neutral Delay ◽  
Competitive System ◽  
Competitive Systems ◽  
Local Asymptotic Stability ◽  
Discrete Delays

AbstractThe qualitative behavior of positive solutions of the neutral-delay two-species Lotka-Volterra competitive system with several discrete delays is investigated. Sufficient conditions are obtained for the local asymptotic stability of the positive steady state. In fact, some of these sufficient conditions are also necessary except at those critical values. Results on the oscillatory and non-oscillatory characteristics of the positive solutions are also included.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

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