Oscillations in Singularly Perturbed Delay Equations

Ivanov, Anatoli F.; Sharkovsky, A. N.

doi:10.1007/978-3-642-61243-5_5

Cited by 68 publications

(74 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Its dynamics have been extensively and intensively studied for the case of g(u) = αu, where α > 0; see, e.g., [14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Section: Remark 13mentioning

confidence: 99%

Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

Yuan

Chen

2018

Adv Differ Equ

View full text Add to dashboard Cite

This paper deals with a large class of non-monotone time-delayed reaction-diffusion equations in which the reaction term can be spatially nonlocal. Nonexistence, existence, uniqueness and global attractivity of positive equilibriums to the equation are addressed. In particular, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractiveness of the unique positive equilibrium. Two examples are given to illustrate the obtained results.

show abstract

“…Its dynamics have been extensively and intensively studied for the case of g(u) = αu, where α > 0; see, e.g., [14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Section: Remark 13mentioning

confidence: 99%

Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

Yuan

Chen

2018

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…In the remainder of this section we study the asymptotic behavior of the system in (10) under a set of reasonable assumptions on function q. In particular, we adopt the gamma kernels in (11) and apply the MacDonald's linear chain technique to derive sufficient conditions for convergence of the dual algorithm [20].…”

Section: Single-resource Casementioning

confidence: 99%

“…We note that a similar approach has been used in [10,11,21] in the past to study the behavior of delay-differential systems. We apply our results to derive (necessary and) sufficient conditions with the same utility and resource price functions studied in [29,32].…”

mentioning

confidence: 99%

Convergence of Dual Algorithm with Arbitrary Communication Delays

Ranjan

2008

SIAM J. Appl. Math.

View full text Add to dashboard Cite

We study the issue of convergence of user rates and resource prices under a family of rate control schemes called dual algorithms with arbitrary communication delays. We first consider a case where a single resource is shared by many users. Then, we study a general network shared by heterogeneous users and derive sufficient conditions for convergence. We show that in the case of a single user utilizing a single resource, our condition is also necessary. Using our results we derive a sufficient condition for convergence with a family of popular utility and resource price functions. We present numerical examples to validate our analysis.

show abstract

“…Then, we apply these results to the system with the user utility and resource price functions of (6) and (8), respectively, in Section VII.…”

Section: Network Model With Delaysmentioning

confidence: 99%

“…Our analysis is based on the invariance-based global stability results for nonlinear delay differential equations [7], [8], [18]. This kind of global stability results are different from those based on Lyapunov or Razumikhin theorems for delay differential equations used in [2], [3], [19], and [28] or from passivity approach [29], and our set up also hints at the structure of emerging periodic orbits (such as their periodicity and amplitude) in the case of loss of stability.…”

Section: Introductionmentioning

confidence: 99%

Global stability conditions for rate control with arbitrary communication delays

La¹,

Ranjan²,

Abed³

2004

2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)

View full text Add to dashboard Cite

Abstract-We analyze Kelly's optimization framework for a rate allocation problem in communication networks and provide stability conditions with arbitrary fixed communication delays. We demonstrate the existence of a fundamental tradeoff between users' price elasticity of demand and the responsiveness of resource through a choice of price function. We also show that the stability of the system can be studied by looking at a much simpler discrete time system that emerges from the underlying market structure of the rate control system with a homogeneous delay. We study the effects of nonresponsive traffic on system stability and show that the presence of nonresponsive traffic enhances the stability of system. We also investigate the system behavior beyond stable regime.

show abstract

Oscillations in Singularly Perturbed Delay Equations

Cited by 68 publications

References 36 publications

Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

Convergence of Dual Algorithm with Arbitrary Communication Delays

Global stability conditions for rate control with arbitrary communication delays

Contact Info

Product

Resources

About