Asymptotic stability of singularly perturbed systems which have marginally stable boundary layer systems

2005 ◽  
pp. 115-126
Author(s):  
Martin Corless
Keyword(s):  
Boundary Layer ◽  
Asymptotic Stability ◽  
Stable Boundary Layer ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Stable Boundary ◽  
Perturbed Systems

scholarly journals The Contrast Structures for a Class of Singularly Perturbed Systems with Heteroclinic Orbits

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Han Xu ◽  
Yinlai Jin
Keyword(s):  
Boundary Layer ◽  
Equilibrium Points ◽  
Singularly Perturbed ◽  
Heteroclinic Orbits ◽  
Singularly Perturbed Systems ◽  
Smooth Solutions ◽  
Singularly Perturbed Problems ◽  
Perturbed Systems ◽  
Contrast Structures ◽  
Smooth Connection

Singularly perturbed problems are often used as the models of ecology and epidemiology. In this paper, a class of semilinear singularly perturbed systems with contrast structures are discussed. Firstly, we verify the existence of heteroclinic orbits connecting two equilibrium points about the associated systems for contrast structures in the corresponding phase space. Secondly, the asymptotic solutions of the contrast structures by the method of boundary layer functions and smooth connection are constructed. Finally, the uniform validity of the asymptotic expansion is defined and the existence of the smooth solutions is proved.

Stability Analysis for a Class of Singularly Perturbed Systems With Multiple Time Delays

2004 ◽  
Vol 126 (3) ◽  
pp. 462-466 ◽  
Author(s):  
Shing-Tai Pan ◽  
Ching-Fa Chen ◽  
Jer-Guang Hsieh
Keyword(s):  
Asymptotic Stability ◽  
Time Delays ◽  
Original System ◽  
Singularly Perturbed ◽  
Multiple Time ◽  
Singularly Perturbed Systems ◽  
Fast Subsystem ◽  
Multiple Time Delays ◽  
Perturbed Systems ◽  
Slow Subsystem

The paper is to investigate the asymptotic stability for a general class of linear time-invariant singularly perturbed systems with multiple non-commensurate time delays. It is a common practice to investigate the asymptotic stability of the original system by establishing that of its slow subsystem and fast subsystem. A frequency-domain approach is first presented to determine a sufficient condition for the asymptotic stability of the slow subsystem (reduced-order model), which is a singular system with multiple time delays, and the fast subsystem. Two delay-dependent criteria, ε-dependent and ε-independent, are then proposed in terms of the H∞-norm for the asymptotic stability of the original system. Furthermore, a simple estimate of an upper bound ε* of singular perturbation parameter ε is proposed so that the original system is asymptotically stable for any ε∈0,ε*. Two numerical examples are provided to illustrate the use of our main results.

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