A.V. Lasunskii. Refinement of estimates for the Lyapunov exponents of a class of linear nonautonomous system of difference equations ... P. 84-90

We obtain an estimate for the norm of an $n$th-order square matrix $A^{t}$: $$ \|A^{t}\|\leq \sum^{n-1}_{k=0}C^{k}_{t}\gamma^{t-k}(\gamma+\|A\|)^{k},\quad t\geq n-1, $$ where $C^{k}_{t}$ are the binomial coefficients, $\gamma=\max\limits_{i}|\lambda_{i}|$, and $\lambda_{i}$ are the eigenvalues of~$A$. Based on this estimate and using the freezing method, we improve the constants in the upper and lower estimates for the highest and lowest exponents, respectively, of the system $ x(t+1)=A(t)x(t),\ x\in \mathbb R^{n},\ t\in \mathbb Z^{+}, $ with a completely bounded matrix $A(t)$. It is assumed that the matrices $A(t)$ and $A^{-1} (t)$ satisfy the inequalities $ \|A(t)-A(s)\|\leq\delta|t-s|^{\alpha},\ \|A^{-1}(t)-A^{-1}(s)\|\leq\delta|t-s|^{\alpha} $ with some constants $0<\alpha\leq 1$ and $\delta>0$ for any $t,s\in\mathbb Z^{+}$. We give an example showing that the constants $\gamma$ and $\delta$ are generally related.

Keywords: estimates for Lyapunov exponents, freezing method for discrete systems

Received April 28, 2020

Revised May 16, 2020

Accepted Juny 30, 2020

Alexandr Vasil’evich Lasunskii, Dr. Phys.-Math. Sci., Yaroslav-the-Vise Novgorod State University, Veliky Novgorod, 173003 Russia, e-mail: Alexandr.Lasunsky@novsu.ru

REFERENCES

1.   Kuznetsov N.V., Alexeeva T.A., Leonov G.A. Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations. Nonlinear Dyn., 2016, vol. 85, pp. 195–201. doi: 10.1007/s11071-016-2678-4 

2.   Czornik A., Nawrat A. On new estimates for Lyapunov exponents of discrete time varying linear systems. Automatica, 2010, vol. 46, no. 4, pp. 775—778 . doi: 10.1016/j.automatica.2010.01.014 

3.   Czornik A., Mokry P., Nawrat A. On the sigma exponent of discrete linear systems. IEEE Transactions on Automatic Control, 2010, vol. 55, no. 6, pp. 1511–1515 . doi: 10.1109/TAC.2010.2045699 

4.   Czornik A., Nawrat A., Niezabitowski M. On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations. Dynamical Systems, 2013, vol. 28, no. 4, pp. 473–483. doi: 10.1080/14689367.2012.748718 

5.   Sergeev I.N. Definition of characteristic frequencies of a linear equation. Diff. Eq., 2004, vol. 40, no. 11, p. 1573 (in Russian) .

6.   Sergeev I.N. Oscillation and wandering characteristics of solutions of a linear differential system.Izvestiya: Mathematics, 2012, vol.76, no.1, pp. 139–162. doi: 10.1070/IM2012v076n01ABEH002578 

7.   Lasunskii A.V. Estimates for solutions of Linear and quasilinear systems in the nonautonomous case. Diff. Eq., 2016, vol. 52, no. 2, pp. 177–185. doi: 10.1134/S001226611602004X 

8.   Zamkovaya L.D. On a method of “Freezing” for discrete systems. Differ. Uravn., 1980, vol. 16, no. 4, pp. 697–704 (in Russian).

9.   Zamkovaya L.D. Estimates of exponents of exponential growth of solutions of some systems. Differ. Uravn., 1988, vol. 24, no. 11, pp. 2008–2010 (in Russian).

10.   Gel’fand I.M., Shilov G.E. Generalized functions, vol. 3: Theory of differential equations. Providence: AMS Chelsea Publ., 1967, 222 p. ISBN: 978-1-4704-2661-3 . Original Russian text published in Gel’fand I.M., Shilov G.E. Nekotorye voprosy teorii differentsial’nykh uravnenii. Moscow: Fizmatgiz Publ., 1958, 274 p.

11.   Bylov B.F., Vinograd R.E., Grobman D.M., and Nemytskii V.V. Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti [Theory of Lyapunov exponents and its application to problems of stability]. Moscow: Nauka Publ., 1966, 576 p.

12.   Izobov N.A. Vvedenie v teoriyu pokazatelei Lyapunova [Introduction to the theory of Lyapunov exponents]. Minsk: BGU Publ., 2006, 319 p. ISBN: 985-485-515-5 .

13.   Lasunskii A.V. Stability and eigenvalues of linear nonautonomous systems of difference and differential equations. Matematika v Vysshem Obrazovanii, 2010, no. 8, pp. 37–40 (in Russian).

Cite this article as: A.V. Lasunskii. Refinement of estimates for the Lyapunov exponents of a class of linear nonautonomous system of difference equations, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 84–90.