| Title:
|
Convergence to equilibria in a differential equation with small delay (English) |
| Author:
|
Pituk, Mihály |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
293-299 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Consider the delay differential equation \[ \dot{x}(t)=g(x(t),x(t-r)), \qquad \mathrm{(1)}\] where $r>0$ is a constant and $g\:\mathbb{R}^2\rightarrow \mathbb{R}$ is Lipschitzian. It is shown that if $r$ is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay. (English) |
| Keyword:
|
delay differential equation |
| Keyword:
|
equilibrium |
| Keyword:
|
convergence |
| MSC:
|
34K12 |
| MSC:
|
34K25 |
| idZBL:
|
Zbl 1016.34076 |
| idMR:
|
MR1981534 |
| DOI:
|
10.21136/MB.2002.134154 |
| . |
| Date available:
|
2012-10-05T13:03:13Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134154 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
[8] T. Krisztin, H.-O. Walther, J. Wu: Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Positive Feedback.Fields Institute Monograph Series, Vol. 11, Amer. Math. Soc., Providence, RI, 1999. MR 1719128 |
| Reference:
|
[9] M. Pituk: Convergence to equilibria in scalar non-quasi-monotone functional differential equations.In preparation. |
| Reference:
|
[10] Yu. A. Ryabov: Certain asymptotic properties of linear systems with small time lag.Trudy Sem. Teor. Differencial. Uravnenii s Otklon. Argumentom Univ. Druzby Narodov Patrica Lumumby 3 (1965), 153–164. (Russian) MR 0211010 |
| Reference:
|
[11] H. L. Smith: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems.Amer. Math. Soc., Providence, RI, 1995. Zbl 0821.34003, MR 1319817 |
| Reference:
|
[12] H. L. Smith, H. Thieme: Monotone semiflows in scalar non-quasi-monotone functional differential equations.J. Math. Anal. Appl. 150 (1990), 289–306. MR 1067429, 10.1016/0022-247X(90)90105-O |
| . |
