Abstract
This article is devoted to the qualitative theory of the stability of the motion of dynamical systems on a metric space. A number of stably similar properties of closed invariant sets are presented, on the basis of which equivalent criteria for their asymptotic stability in semidynamical systems are formulated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. M. Lyapunov, The General Problem of the Stability of Motion (Gostekhizdat, Moscow, 1950; Academic Press, New York, 1966).
R. É. Vinograd, “Inapplicability of the method of characteristic exponents to the study of non-linear differential equations,” Mat. Sb. (N.S.) 41, 431–438 (1957).
H. A. Antosiewicz, “A survey of Liapunov’s second metod,” in Contributions to the Theory of Non-Linear Oscillation, Ed. by S. Lefschetz, Annals of Mathematical Studies (Princeton Univ. Press, 1958), Vol. 4, pp. 141–166.
B. P. Mendelson, “On unstable attractors,” Boletian Math. Soc. Mex. 5 (2), 270–276 (1960).
S. Lefschetz, “Liapunov and stability in dynamical systems,” Boletian Math. Soc. Mex. 3 (2), 25–39 1958.
N. P. Bhatia, “Weak attractors in dynamical systems,” Boletian Math. Soc. Mex. 11, 56–64 1966.
N. P. Bhatia and G. Szeg, Stability Theory of Dynamical Systems (Springer, Berlin, 1970).
V. I. Zubov, Stability of Motion: Lyapunov Methods and Their Application, 2nd ed. (Vysshaya Shkola, Moscow, 1984).
M. S. Izman, “Stability and attraction sets in disperse dynamical systems,” Mat. Iissled. 3 (3), 60–78 (1968).
B. S. Kalitin, Qualitative Theory of Stability of Motion of Dynamical Systems (Beloruss. Gos. Univ., Minsk, 2002).
J. H. Arredondo and P. Seibert, “On a characterization of asymptotic stability,” Aportaciones Mat. Ser. Comunicationes 29, 11–16 (2001).
B. S. Kalitine, “About asymptotic stability in semidynamical systems,” Vestn. Beloruss. Gos. Univ. Ser. 1: Fiz., Mat., Inf., No. 1, 114–119 (2016).
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math Surv. Monographs, Vol. 25 (Am. Math. Soc., Malabar, Fla., 1988).
O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations (Cambridge Univ. Press, Cambridge, 1991).
S. H. Saperstone, Semidynamical Systems in Infinite Dimentional Space, Applied Mathematical Sciences, Vol. 37 (Springer, New York, 1981). https://doi.org/10.1007/978-1-4612-5977-0
K. S. Sibirskii and A. S. Shube, Semidynamical Systems (Shtiintsa, Chisinau, 1987).
G. D. Birkhoff, Dynamical Systems (Am. Math. Soc., 1927).
B. S. Kalitin, “Instability of closed invariant sets of semidynamical systems: Method of sign-constant Lyapunov functions,” Math. Notes 85, 374–384 (2009). https://doi.org/10.1134/S0001434609030080
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by E. Chernokozhin
About this article
Cite this article
Kalitine, B.S. On Criteria for Asymptotic Stability. Russ Math. 66, 26–32 (2022). https://doi.org/10.3103/S1066369X22090031
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X22090031