Lyapunov regularity via singular values. (English) Zbl 1384.34018
The authors describe various relations between Lyapunov regularity (for linear nonautonomous dynamics), expressed in terms of Lyapunov exponents and exponential growth rates of the singular values.
In particular, they obtain the following: (i) Sequences of matrices for which any given value of the Lyapunov exponent and of the growth rate of the singular values are attained. (ii) Upper bounds for the values of the Lyapunov exponent in terms of the exponential growth rate of the singular values. (iii) A structure of Oseledets type for any nonregular tempered dynamics that is analogous to that in the multiplicative ergodic theorem. (iv) A simple proof of various characterizations of Lyapunov regularity as well as a new characterization.
Both discrete and continuous cases are considered.
In particular, they obtain the following: (i) Sequences of matrices for which any given value of the Lyapunov exponent and of the growth rate of the singular values are attained. (ii) Upper bounds for the values of the Lyapunov exponent in terms of the exponential growth rate of the singular values. (iii) A structure of Oseledets type for any nonregular tempered dynamics that is analogous to that in the multiplicative ergodic theorem. (iv) A simple proof of various characterizations of Lyapunov regularity as well as a new characterization.
Both discrete and continuous cases are considered.
Reviewer: Pavel Rehak (Brno)
MSC:
| 34A30 | Linear ordinary differential equations and systems |
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |
| 37H15 | Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents |
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