Regularity and stability sets for families of sequences of matrices. (English) Zbl 1456.37030
Authors’ abstract: We consider the notions of Lyapunov regularity and of Lyapunov stability and asymptotic
stability for a dynamics defined by a continuous 1-parameter family of sequences of matrices.
In particular, we identify all classes of sets that can be the regularity set, the stability set and
the asymptotic stability set of any such family. Moreover, we construct explicitly families of
sequences of matrices whose regularity set, stability set or asymptotic stability set is a given
set in those classes.
Reviewer: Hongying Shu (Xi’an)
MSC:
| 37C75 | Stability theory for smooth dynamical systems |
| 37D99 | Dynamical systems with hyperbolic behavior |
References:
| [1] | Adrianova, L., Introduction to Linear Systems of Differential Equations, Translations of Mathematical Monographs (1995), Providence: American Mathematical Society, Providence · Zbl 0844.34001 |
| [2] | Barabanov, E., On the improperness sets of families of linear differential systems, Differ. Eq., 45, 1087-1104 (2009) · Zbl 1192.34061 · doi:10.1134/S0012266109080011 |
| [3] | Barabanov, E., Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I, Differ. Eq., 46, 613-627 (2010) · Zbl 1214.34043 · doi:10.1134/S0012266110050010 |
| [4] | Barabanov, E., Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: II, Differ. Eq., 46, 798-807 (2010) · Zbl 1214.34044 · doi:10.1134/S0012266110060042 |
| [5] | Barabanov, E.; Kasabutskii, A., Structure of nonasymptotic stability sets of families of linear differential systems with a multiplying parameter, Differ. Eq., 47, 153-165 (2011) · Zbl 1235.34163 · doi:10.1134/S0012266111020017 |
| [6] | Barreira, L., Lyapunov Exponents (2017), Basel: Birkhäuser, Basel · Zbl 1407.37001 |
| [7] | Barreira, L.; Pesin, Y., Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series (2002), Providence: American Mathematical Society, Providence |
| [8] | Barreira, L.; Pesin, Y., Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications (2007), Cambridge: Cambridge University Press, Cambridge · Zbl 1144.37002 |
| [9] | Barreira, L.; Valls, C., Lyapunov regularity via singular values, Trans. Am. Math. Soc., 369, 8409-8436 (2017) · Zbl 1384.34018 · doi:10.1090/tran/6910 |
| [10] | Bhatia, N.; Szegö, G., Stability Theory of Dynamical Systems, Grundlehren der mathematischen Wissenschaften (1970), Berlin: Springer, Berlin · Zbl 0213.10904 |
| [11] | Bonatti, C.; Díaz, L.; Viana, M., Dynamics beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences (2005), Berlin: Springer, Berlin · Zbl 1060.37020 |
| [12] | Czornik, A., Perturbation Theory for Lyapunov Exponents of Discrete Linear Systems (2012), Kraków: Wydawnictwa AGH, Kraków |
| [13] | Hahn, W., Stability of Motion, Grundlehren der mathematischen Wissenschaften (1967), Berlin: Springer, Berlin · Zbl 0189.38503 |
| [14] | Hausdorff, F., Set Theory (1962), White River Junction: Chelsea Publishing Co., White River Junction |
| [15] | Izobov, N., Lyapunov Exponents and Stability, Stability Oscillations and Optimization of Systems (2012), Cambridge: Cambridge Scientific Publishers, Cambridge · Zbl 1402.93011 |
| [16] | LaSalle, J.; Lefschetz, S., Stability by Liapunov’s Direct Method, with Applications, Mathematics in Science and Engineering (1961), Cambridge: Academic Press, Cambridge · Zbl 0098.06102 |
| [17] | Lyapunov, A., The General Problem of the Stability of Motion (1992), Abingdon: Taylor & Francis, Abingdon · Zbl 0786.70001 |
| [18] | Oseledets, V., A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19, 197-221 (1968) · Zbl 0236.93034 |
| [19] | Pesin, Y., Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40, 1261-1305 (1976) · Zbl 0383.58012 · doi:10.1070/IM1976v010n06ABEH001835 |
| [20] | Pesin, Y., Characteristic Ljapunov exponents, and smooth ergodic theory, Rus. Math. Surv., 32, 55-114 (1977) · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639 |
| [21] | Pesin, Y., Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics (2004), Zurich: European Mathematical Society, Zurich · Zbl 1098.37024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
