Stochastic storage networks: stationarity and the feedforward case

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

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Stability of the Difference Schemes for the Equations of Weakly Compressible Liquid

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2007-0012 ◽

2007 ◽

Vol 7 (3) ◽

pp. 208-220 ◽

Cited By ~ 2

Author(s):

P. Matus ◽

O. Korolyova ◽

M. Chuiko

Keyword(s):

Initial Data ◽

Difference Scheme ◽

Initial Conditions ◽

Compressible Liquid ◽

Stability Conditions ◽

Differential Problem ◽

Weakly Compressible ◽

Linear Problems ◽

The Difference ◽

The Stability

Abstract A priory estimates of the stability in the sense of the initial data of the difference scheme approximating weakly compressible liquid equations in the Riemann invariants have been obtained. These estimates have been proved without any assumptions about the properties of the solution of the differential problem and depend only on the behavior of the initial conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for a finite instant of time t≤t_0. In particular, this is confirmed by the fact, that nonfulfilment of these stability conditions lead to the appearance of supersonic flows or domains with large gradients. The questions of uniqueness and convergence of the difference solution are considered also. The results of the computating experiment confirming the theoretical conclusions are given.

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Stability and structural properties of stochastic storage networks

Journal of Applied Probability ◽

10.1017/s0021900200100567 ◽

1996 ◽

Vol 33 (04) ◽

pp. 1169-1180 ◽

Cited By ~ 5

Author(s):

Offer Kella ◽

Ward Whitt

Keyword(s):

Structural Properties ◽

Stability Condition ◽

General Model ◽

Initial Conditions ◽

Internal Flows ◽

Stationary Increments ◽

Fluid Network ◽

Storage Networks ◽

Stochastic Fluid

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

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Stability and structural properties of stochastic storage networks

Journal of Applied Probability ◽

10.2307/3214994 ◽

1996 ◽

Vol 33 (4) ◽

pp. 1169-1180 ◽

Cited By ~ 33

Author(s):

Offer Kella ◽

Ward Whitt

Keyword(s):

Structural Properties ◽

Stability Condition ◽

General Model ◽

Initial Conditions ◽

Internal Flows ◽

Stationary Increments ◽

Fluid Network ◽

Storage Networks ◽

Stochastic Fluid

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

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Initial-Condition Sensitivities and the Predictability of Downslope Winds

Journal of the Atmospheric Sciences ◽

10.1175/2009jas3023.1 ◽

2009 ◽

Vol 66 (11) ◽

pp. 3401-3418 ◽

Cited By ~ 48

Author(s):

Patrick A. Reinecke ◽

Dale R. Durran

Keyword(s):

Wave Breaking ◽

Initial Conditions ◽

Static Stability ◽

Synoptic Scale ◽

Wind Speeds ◽

Downslope Winds ◽

First Case ◽

Downslope Wind ◽

The Difference ◽

Strong Winds

Abstract The sensitivity of downslope wind forecasts to small changes in initial conditions is explored by using 70-member ensemble simulations of two prototypical windstorms observed during the Terrain-Induced Rotor Experiment (T-REX). The 10 weakest and 10 strongest ensemble members are composited and compared for each event. In the first case, the 6-h ensemble-mean forecast shows a large-amplitude breaking mountain wave and severe downslope winds. Nevertheless, the forecasts are very sensitive to the initial conditions because the difference in the downslope wind speeds predicted by the strong- and weak-member composites grows to larger than 28 m s−1 over the 6-h forecast. The structure of the synoptic-scale flow one hour prior to the windstorm and during the windstorm is very similar in both the weak- and strong-member composites. Wave breaking is not a significant factor in the second case, in which the strong winds are generated by a layer of high static stability flowing beneath a layer of weaker mid- and upper-tropospheric stability. In this case, the sensitivity to initial conditions is weaker but still significant. The difference in downslope wind speeds between the weak- and strong-member composites grows to 22 m s−1 over 12 h. During and one hour before the windstorm, the synoptic-scale flow exhibits appreciable differences between the strong- and weak-member composites. Although this case appears to be more predictable than the wave-breaking event, neither case suggests that much confidence should be placed in the intensity of downslope winds forecast 12 or more hours in advance.

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Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i230081 ◽

2019 ◽

pp. 1-12

Author(s):

Abdualrazaq Sanbo ◽

Elsayed M. Elsayed ◽

Faris Alzahrani

Keyword(s):

Difference Equation ◽

Difference Equations ◽

Analytical Study ◽

Initial Conditions ◽

Specific Form ◽

Positive Real ◽

Rational Difference Equations ◽

Special Cases ◽

The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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Global Behavior of the Difference Equationxn+1=xn-1g(xn)

Abstract and Applied Analysis ◽

10.1155/2014/705893 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Hongjian Xi ◽

Taixiang Sun ◽

Bin Qin ◽

Hui Wu

Keyword(s):

Continuous Function ◽

Difference Equation ◽

Positive Solution ◽

Positive Solutions ◽

Initial Conditions ◽

Global Behavior ◽

Initial Values ◽

The Difference ◽

Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

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Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4039626 ◽

2018 ◽

Vol 13 (5) ◽

Cited By ~ 6

Author(s):

Eric Donald Dongmo ◽

Kayode Stephen Ojo ◽

Paul Woafo ◽

Abdulahi Ndzi Njah

Keyword(s):

Chaotic System ◽

Initial Conditions ◽

Lyapunov Stability Theory ◽

The State ◽

State Variables ◽

State Variable ◽

New Type ◽

The Difference ◽

Synchronization Scheme ◽

Hyperchaotic Systems

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

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Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421502333 ◽

2021 ◽

Vol 31 (15) ◽

Author(s):

Penghe Ge ◽

Hongjun Cao

Keyword(s):

Neuron Model ◽

Initial Conditions ◽

Stability Conditions ◽

Two Dimensional ◽

Bifurcation Diagrams ◽

Fast Subsystem ◽

One Dimensional ◽

Sensitive Dependence ◽

The Stability ◽

Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

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COMPARATIVE ANALYSIS OF THE RESULTS OF MODELING THE DEPRECIATION OF SMALL ARMS WHEN FIRED IN QUASI-STATIC AND DYNAMIC PRODUCTIONS

Fundamental and Applied Problems of Engineering and Technology ◽

10.33979/2073-7408-2020-344-6-24-28 ◽

2020 ◽

Vol 6 ◽

pp. 24-28

Author(s):

V.L. BARANOV ◽

A.S. LEVIN

Keyword(s):

Comparative Analysis ◽

Elastic Element ◽

Shock Absorber ◽

Initial Conditions ◽

Small Arms ◽

Dynamic Formulation ◽

The Difference

Various variants of the initial conditions for calculating the dynamics of the elastic element of the shock absorber are considered. A comparative analysis of the influence of initial conditions on the difference in calculations in the quasi-static and dynamic formulation is carried out.

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