• Title, Summary, Keyword: global stability

### BEHAVIOR OF POSITIVE SOLUTIONS OF A DIFFERENCE EQUATION

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.217-230
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• 2017
• In this paper we deal with the difference equation $$y_{n+1}=\frac{ay_{n-1}}{by_ny_{n-1}+cy_{n-1}y_{n-2}+d}$$, $$n{\in}\mathbb{N}_0$$, where the coefficients a, b, c, d are positive real numbers and the initial conditions $y_{-2}$, $y_{-1}$, $y_0$ are nonnegative real numbers. Here, we investigate global asymptotic stability, periodicity, boundedness and oscillation of positive solutions of the above equation.

### THE DYNAMICS OF POSITIVE SOLUTIONS OF A HIGHER ORDER FRACTIONAL DIFFERENCE EQUATION WITH ARBITRARY POWERS

• GUMUS, MEHMET;SOYKAN, YUKSEL
• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.267-276
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• 2017
• The purpose of this paper is to investigate the local asymptotic stability of equilibria, the periodic nature of solutions, the existence of unbounded solutions and the global behavior of solutions of the fractional difference equation $$x_{n+1}=\frac{^{{\alpha}x}n-1(k+1)}{{\beta}+{\gamma}x^p_{n-k}x^q_{n-(k+2)}}$$, $$n=0,1,{\dots}$$ where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-(k+2)}$,$x_{-(k+1)}$, ${\dots}$, $x_{-1}$, $x_0{\in}\mathb{R}^+$.

### A NONSTANDARD FINITE DIFFERENCE METHOD APPLIED TO A MATHEMATICAL CHOLERA MODEL

• Liao, Shu;Yang, Weiming
• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1893-1912
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• 2017
• In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.

### DENSITY DEPENDENT MORTALITY OF INTERMEDIATE PREDATOR CONTROLS CHAOS-CONCLUSION DRAWN FROM A TRI-TROPHIC FOOD CHAIN

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.179-199
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• 2018
• The paper explores a tri-trophic food chain model with density dependent mortality of intermediate predator. To analyze this aspect, we have worked out the local stability of different equilibrium points. We have also derived the conditions for global stability of interior equilibrium point and conditions for persistence of model system. To observe the global behaviour of the system, we performed extensive numerical simulations. Our simulation results reveal that chaotic dynamics is produced for increasing value of half-saturation constant. We have also observed trajectory motions around different equilibrium points. It is noticed that chaotic dynamics has been controlled by increasing value of density dependent mortality parameter. So, we conclude that the density dependent mortality parameter can be used to control chaotic dynamics. We also applied basic tools of nonlinear dynamics such as Poincare section and Lyapunov exponent to investigate chaotic behaviour of the system.

### DYNAMICAL BEHAVIOUR OF A DRINKING EPIDEMIC MODEL

• Sharma, Swarnali;Samanta, G.P.
• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.747-767
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• 2013
• In this paper we have constructed a mathematical model of alcohol abuse which consists of four compartments corresponding to four population classes, namely, moderate and occasional drinkers, heavy drinkers, drinkers in treatment and temporarily recovered class. Basic reproduction number $R_0$ has been determined and sensitivity analysis of $R_0$ indicates that ${\beta}1$ (the transmission coefficient from moderate and occasional drinker to heavy drinker) is the most useful parameter for preventing drinking habit. Stability analysis of the model is made using the basic reproduction number. The model is locally asymptotically stable at disease free or problem free equilibrium (DFE) $E_0$ when $R_0&lt;1$. It is found that, when $R_0=1$, a backward bifurcation can occur and when $R_0&gt;1$, the endemic equilibrium $E^*$ becomes stable. Further analysis gives the global asymptotic stability of DFE under some conditions. Our important analytical findings are illustrated through computer simulation. Epidemiological implications of our analytical findings are addressed critically.

### An Extended Model Evaluation Method under Uncertainty in Hydrologic Modeling

• Lee, Giha;Youn, Sangkuk;Kim, Yeonsu
• Journal of the Korean GEO-environmental Society
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• v.16 no.5
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• pp.13-25
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• 2015
• This paper proposes an extended model evaluation method that considers not only the model performance but also the model structure and parameter uncertainties in hydrologic modeling. A simple reservoir model (SFM) and distributed kinematic wave models (KWMSS1 and KWMSS2 using topography from 250-m, 500-m, and 1-km digital elevation models) were developed and assessed by three evaluative criteria for model performance, model structural stability, and parameter identifiability. All the models provided acceptable performance in terms of a global response, but the simpler SFM and KWMSS1 could not accurately represent the local behaviors of hydrographs. Moreover, SFM and KWMSS1 were structurally unstable; their performance was sensitive to the applied objective functions. On the other hand, the most sophisticated model, KWMSS2, performed well, satisfying both global and local behaviors. KMSS2 also showed good structural stability, reproducing hydrographs regardless of the applied objective functions; however, superior parameter identifiability was not guaranteed. A number of parameter sets could result in indistinguishable hydrographs. This result indicates that while making hydrologic models complex increases its performance accuracy and reduces its structural uncertainty, the model is likely to suffer from parameter uncertainty.

### Bicriteria optimal design of open cross sections of cold-formed thin-walled beams

• Ostwald, M.;Magnucki, K.;Rodak, M.
• Steel and Composite Structures
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• v.7 no.1
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• pp.53-70
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• 2007
• This paper presents a analysis of the problem of optimal design of the beams with two I-type cross section shapes. These types of beams are simply supported and subject to pure bending. The strength and stability conditions were formulated and analytically solved in the form of mathematical equations. Both global and selected types of local stability forms were taken into account. The optimization problem was defined as bicriteria. The cross section area of the beam is the first objective function, while the deflection of the beam is the second. The geometric parameters of cross section were selected as the design variables. The set of constraints includes global and local stability conditions, the strength condition, and technological and constructional requirements in the form of geometric relations. The optimization problem was formulated and solved with the help of the Pareto concept of optimality. During the numerical calculations a set of optimal compromise solutions was generated. The numerical procedures include discrete and continuous sets of the design variables. Results of numerical analysis are presented in the form of tables, cross section outlines and diagrams. Results are discussed at the end of the work. These results may be useful for designers in optimal designing of thin-walled beams, increasing information required in the decision-making procedure.

### A Comparative Study of Subsea Pipeline Global Buckling Control Method (해저 파이프라인의 전체 좌굴 제어 방법 비교)

• Kim, Koo;Kim, Do-Kyun;Choi, Han-Suk;Park, Kyu-Sik
• Journal of the Korean Society for Advanced Composite Structures
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• v.6 no.1
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• pp.51-58
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• 2015
• Global buckling is a bending of pipeline and it occurs when the stability of pipeline is distributed by excessive axial force. Subesea pipeline is subjected to axial force induced by temperature and pressure from well and resulting phenomena should be controlled in appropriate manner. Global buckling of subsea pipeline is still ongoing research subject and is studied various organization. In this study, various control methods such as buoyancy module, sleeper, and snake lay for global buckling of subsea pipeline were numerically investigated with various design parameters. From the numerical simulation results, the global buckling control method using sleepers shows better results than buoyancy module and snake lay control methods in the sense of combined stress after buckling. Furthermore, the global buckling of full scale pipeline of 80km with uneven seabed profile were successfully managed when the sleeper was installed.

### A Study on Decentralized under Voltage Load Shedding Scheme for Preventing Wide-area Black Out (광역정전 예방을 위한 분산형 부하 제어 방안에 대한 연구)

• Lee, Yun-Hwan
• The Transactions of the Korean Institute of Electrical Engineers P
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• v.63 no.1
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• pp.1-6
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• 2014
• An electric power system sometimes fails because of disturbances that occur unexpectedly, such as the uncontrolled loss of load that developed from cascading blackout. Which make stability through a little of under voltage load shedding should work. The development of phasor measurement unit(PMU) makes network supervision possible. The information obtained from PMU is synchronized by global positioning system(GPS). There are many real-time algorithms which are monitoring the voltage stability. This paper presents the study on the VILS(Voltage Instability Load Shedding) using PMU data. This algorithm computes Voltage Stability Margin Index(VSMI) continuously to track the voltage stability margin at local bus level. The VSMI is expressed as active and reactive power. The VSMI is used as an criterion for load shedding. In order to examine the algorithm is effective, applied to KEPCO system.

### On the local stability condition in the planar beam finite element

• Planinc, Igor;Saje, Miran;Cas, Bojan
• Structural Engineering and Mechanics
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• v.12 no.5
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• pp.507-526
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• 2001
• In standard finite element algorithms, the local stability conditions are not accounted for in the formulation of the tangent stiffness matrix. As a result, the loss of the local stability is not adequately related to the onset of the global instability. The phenomenon typically arises with material-type localizations, such as shear bands and plastic hinges. This paper addresses the problem in the context of the planar, finite-strain, rate-independent, materially non-linear beam theory, although the proposed technology is in principle not limited to beam structures. A weak formulation of Reissner's finite-strain beam theory is first presented, where the pseudocurvature of the deformed axis is the only unknown function. We further derive the local stability conditions for the large deformation case, and suggest various possible combinations of the interpolation and numerical integration schemes that trigger the simultaneous loss of the local and global instabilities of a statically determined beam. For practical applications, we advice on a procedure that uses a special numerical integration rule, where interpolation nodes and integration points are equal in number, but not in locations, except for the point of the local instability, where the interpolation node and the integration point coalesce. Provided that the point of instability is an end-point of the beam-a condition often met in engineering practice-the procedure simplifies substantially; one of such algorithms uses the combination of the Lagrangian interpolation and Lobatto's integration. The present paper uses the Galerkin finite element discretization, but a conceptually similar technology could be extended to other discretization methods.