| Title:
|
Dynamical systems with several equilibria and natural Liapunov functions (English) |
| Author:
|
Răsvan, Vladimir |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
207-215 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Dynamical systems with several equilibria occur in various fields of science and engineering: electrical machines, chemical reactions, economics, biology, neural networks. As pointed out by many researchers, good results on qualitative behaviour of such systems may be obtained if a Liapunov function is available. Fortunately for almost all systems cited above the Liapunov function is associated in a natural way as an energy of a certain kind and it is at least nonincreasing along systems solutions. (English) |
| Keyword:
|
Several equilibria |
| Keyword:
|
qualitative behaviour |
| Keyword:
|
Liapunov function Introduction Dynamical systems with several equilibria occur in various fields of science and engineering: electrical machines |
| Keyword:
|
chemical reactions |
| Keyword:
|
economics |
| Keyword:
|
biology |
| Keyword:
|
neural networks |
| MSC:
|
34C11 |
| MSC:
|
34C99 |
| MSC:
|
34D20 |
| MSC:
|
37-99 |
| idZBL:
|
Zbl 0915.34043 |
| idMR:
|
MR1629709 |
| . |
| Date available:
|
2009-02-17T10:11:17Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107646 |
| . |
| Reference:
|
[1] M. Cohen S. Grossberg: Absolute Stability of Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks.IEEE Trans. on Syst. Man Cybernetics, SMC-13, (1983), 815–826. MR 0730500 |
| Reference:
|
[2] D. A. Frank Kamenetskii: Diffusion and heat transfer in chemical kinetics.(in Russian), Nauka, Moscow 1987. |
| Reference:
|
[3] A. Kh. Gelig G. A. Leonov V. A. Yakubovich: Stability of systems with non-unique equilibria.(in Russian), Nauka, Moscow 1978. MR 0519679 |
| Reference:
|
[4] A. Halanay, Vl. Răsvan: Applications of Liapunov Methods to Stability.Kluwer 1993. |
| Reference:
|
[5] M. Hirsch: Systems of differential equations which are competitive or cooperative. I-Limit sets.SIAM J. Math. Anal, 13, 2, (1982), 167–169. Zbl 0494.34017, MR 0647119 |
| Reference:
|
M. Hirsch: Systems of differential equations which are competitive or cooperative. II-Convergence almost everywhere.SIAM J. Math. Anal, 16, 3, (1985), 423–439. MR 0783970 |
| Reference:
|
[6] M. Hirsch: Stability and convergence in strongly monotone dynamical systems.J. reine angew. Mathem., 383, (1988), 1–53. Zbl 0624.58017, MR 0921986 |
| Reference:
|
[7] R. E. Kalman: Physical and mathematical mechanisms of instability in nonlinear automatic control systems.Trans. ASME, 79 (1957), no. 3 MR 0088420 |
| Reference:
|
[8] S. N. Kružkov A. N. Peregudov: The Cauchy problem for a system of quasilinear parabolic equations of chemical kinetics type.Journ. of Math. Sci., 69, 3, (1994), 1110–1125. MR 1275290 |
| Reference:
|
[9] G. A. Leonov V. Reitmann V. B. Smirnova: Non-local methods for pendulum-like feedback systems.Teubner Verlag 1992. MR 1216519 |
| Reference:
|
[10] J. Moser: On nonoscillating networks.Quart. of Appl. Math., 25 (1967), 1–9. Zbl 0154.34803, MR 0209567 |
| Reference:
|
[11] V. M. Popov: Monotonicity and Mutability.J. of Diff. Eqs., 31 (1979), 337–358. Zbl 0368.45008, MR 0530933 |
| Reference:
|
[12] P. Simon H. Farkas: Globally attractive domains in two-dimensional reversible chemical, dynamical systems.Ann. Univ. Sci. Budapest, Sect. Comp., 15 (1995), 179–200. MR 1428282 |
| Reference:
|
[13] Iu. M. Svirežev: .Appendix to the Russian edition of [14], Nauka, Moscow 1976. |
| Reference:
|
[14] V. Volterra: Leçons sur la théorie mathématique de la lutte pour la vie.Gauthier Villars et Cie, Paris 1931. Zbl 0002.04202 |
| . |
