scholarly journals Well-posedness and Stability of the Repairable System with Three Units and Vacation

2014 ◽  
Vol 2 (1) ◽  
pp. 54-76
Author(s):  
Xiaoshuang Han ◽  
Mingyan Teng ◽  
Ming Fang
Keyword(s):  
Steady State ◽  
Abstract Cauchy Problem ◽  
Steady State Solution ◽  
State Solution ◽  
Repairable System ◽  
Dynamic Solution ◽  
System Operator ◽  
Mean Ergodicity ◽  
System Equations ◽  
The Stability

AbstractThe stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroupT(t) generated by the system operator was also shown through the irreducibility of the semigroup.

The stability of Couette flow of certain anisotropic fluids

1964 ◽  
Vol 60 (4) ◽  
pp. 949-955 ◽  
Author(s):  
F. M. Leslie
Keyword(s):  
Steady State ◽  
Couette Flow ◽  
Steady State Solution ◽  
State Solution ◽  
Rotating Cylinders ◽  
Important Conclusion ◽  
Preferred Direction ◽  
The Stability ◽  
Anisotropic Fluids

AbstractThe stability of the flow between concentric, rotating cylinders is discussed when the gap is small and the cylinders are rotating in the same direction for a class of anisotropic fluids in which the fluid has a preferred direction. An important conclusion of the analysis is that a steady-state solution of the equations has previously been considered unstable on false grounds.

QUALITATIVE ANALYSIS OF A MATHEMATICAL MODEL FOR CAPILLARY FORMATION IN TUMOR ANGIOGENESIS

2003 ◽  
Vol 13 (01) ◽  
pp. 19-33 ◽  
Author(s):  
SERDAL PAMUK
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Qualitative Analysis ◽  
Transition Probability ◽  
Steady State Solution ◽  
State Solution ◽  
Transition Probability Density ◽  
Capillary Formation ◽  
Long Time ◽  
The Stability

Qualitative analysis of a mathematical model for capillary formation is presented under assumptions that enzyme and fibronectin concentrations are in quasi-steady state. The aim of this paper is to prove mathematically that the long-time tendency of endothelial cells will be towards the transition probability density function of enzyme and fibronectin. Endothelial cell steady-state solution is obtained and a numerical simulation is provided to show that there is a close agreement between the steady-state solution obtained analytically and the numerically calculated steady-state of the related initial value problem, which provides strong evidence for the stability of this steady-state.

The stability of localized spikes for the 1-D Brusselator reaction–diffusion model

2013 ◽  
Vol 24 (4) ◽  
pp. 515-564 ◽  
Author(s):  
J. C. TZOU ◽  
Y. NEC ◽  
M. J WARD
Keyword(s):  
Steady State ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
State Solution ◽  
Oscillatory Instability ◽  
Spike Pattern ◽  
Reaction Diffusion Model ◽  
The Stability ◽  
Spike Patterns

In a one-dimensional domain, the stability of localized spike patterns is analysed for two closely related singularly perturbed reaction–diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a non-local eigenvalue problem (NLEP), whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer–Meinhardt and Gray–Scott models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous, rather than synchronous, oscillatory instability of the spike amplitudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike amplitudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike amplitudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full partial differential equations system.

scholarly journals RELIABILITY ANALYSIS OF A SIMPLE REPAIRABLE SYSTEM

2010 ◽  
Vol 52 (2) ◽  
pp. 203-217
Author(s):  
L. N. GUO ◽  
H. B. XU ◽  
C. GAO ◽  
G. T. ZHU
Keyword(s):  
Steady State ◽  
Semigroup Theory ◽  
Point Of View ◽  
Repairable System ◽  
Dynamic Solution ◽  
Profit Analysis ◽  
Laplace Transform Technique ◽  
System Operator ◽  
Common Technique ◽  
Uniqueness And Existence

AbstractWe consider a new kind of simple repairable system consisting of a repairman with multiple delayed-vacation strategy. A common technique in reliability studies is to substitute the steady-state reliability indexes for instantaneous ones because the dynamic solution of the system is difficult or even impossible to obtain. However, this substitution is not always valid. Therefore, it is important to study the existence, uniqueness and expression for the system’s dynamic solution, and to discuss the system’s stability. The purpose of this paper is threefold: to study the uniqueness and existence of the dynamic solution, and its expression, using C0-semigroup theory; to discuss the exponential stability of the system by analysing the spectral distribution and quasi-compactness of the system operator; to derive some reliability indexes of the system from an eigenfunction point of view, which is different from the traditional Laplace transform technique, and present a profit analysis to determine the optimal vacation time in order to achieve the maximum system profit.

The Steady-State Response of the Double Bilinear Hysteretic Model

1965 ◽  
Vol 32 (4) ◽  
pp. 921-925 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Steady State Solution ◽  
Degree Of Freedom ◽  
State Solution ◽  
Steady State Response ◽  
Hysteretic Model ◽  
Jump Phenomenon ◽  
State Response ◽  
The Stability

The steady-state response of a one-degree-of-freedom double bilinear hysteretic model is investigated and it is shown that this model gives rise to the jump phenomenon which is associated with certain nonlinear systems. The stability of the steady-state solution is discussed and it is shown that the model predicts an unbounded resonance for finite excitation.

Bifurcation Analysis in an n-Dimensional Diffusive Competitive Lotka–Volterra System with Time Delay

2015 ◽  
Vol 25 (06) ◽  
pp. 1550089
Author(s):  
Xiaoyuan Chang ◽  
Junjie Wei
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Steady State Solution ◽  
State Solution ◽  
Diffusion System ◽  
Asymptotic Expressions ◽  
The Stability ◽  
System With Time Delay

In this paper, we investigate the stability and Hopf bifurcation of an n-dimensional competitive Lotka–Volterra diffusion system with time delay and homogeneous Dirichlet boundary condition. We first show that there exists a positive nonconstant steady state solution satisfying the given asymptotic expressions and establish the stability of the positive nonconstant steady state solution. Regarding the time delay as a bifurcation parameter, we explore the system that undergoes a Hopf bifurcation near the positive nonconstant steady state solution and derive a calculation method for determining the direction of the Hopf bifurcation. Finally, we cite the stability of a three-dimensional competitive Lotka–Volterra diffusion system with time delay to illustrate our conclusions.

A Nonlinear Theory of Sloshing in Rectangular Tanks

1974 ◽  
Vol 18 (04) ◽  
pp. 224-241 ◽  
Author(s):  
Odd M. Faltinsen
Keyword(s):  
Steady State ◽  
Power Series ◽  
Boundary Value Problem ◽  
Nonlinear Theory ◽  
Reasonable Agreement ◽  
Steady State Solution ◽  
State Solution ◽  
Two Dimensional ◽  
The Stability ◽  
Theory And Experiment

A two-dimensional, rigid, rectangular, open tank without baffles is forced to oscillate harmonically with small amplitudes of sway or roll oscillation in the vicinity of the lowest natural frequency for the fluid inside the tank. The breadth of the tank is 0(1) and the depth of the fluid is either 0(1) or in-finite. The excitation is 0(ε) and the response is 0(ε1/3). A nonlinear, inviscid boundary-value problem of potential flow is formulated and the steady-state solution is found as a power series in ε1/3 correctly to 0(ε). Comparison between theory and experiment shows reasonable agreement. The stability of the steady-state solution has been studied.

Forced Vibration of Tube Support Plates in Power Plant Condensers, With Friction Damping

1977 ◽  
Vol 99 (1) ◽  
pp. 106-114
Author(s):  
R. Pigott
Keyword(s):  
Steady State ◽  
Power Plant ◽  
Forced Vibration ◽  
Steady State Solution ◽  
State Solution ◽  
Friction Damping ◽  
Elastic Plastic ◽  
The Stability

A method is given for analyzing the vibration of tube support plates in power plant condensers. The tubes are not fastened to the support plates and, therefore, their effect is simply that of a frictional restraint which is modeled as a massless elastic-plastic foundation. The technique given is an application of the Kryloff-Bogoliuboff method. The stability of the steady-state solution is also considered and it is shown that “jump” phenonmenon exist for some conditions.

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