Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs
HTML articles powered by AMS MathViewer
- by Janusz Mierczyński PDF
- Proc. Amer. Math. Soc. 143 (2015), 1127-1135 Request permission
Abstract:
For cooperative random linear systems of ordinary differential equations a method is presented of obtaining lower estimates of the top Lyapunov exponent. The proofs are based on applying some polynomial Lyapunov-like function. Known estimates for the dominant eigenvalue of a nonnegative matrix due to G. Frobenius and L. Yu. Kolotilina are shown to be specializations of our results.References
- Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
- M. Benaïm and S. J. Schreiber, Persistence of structured populations in environmental models, Theor. Popul. Biol. 76 (2009), no. 1, 19–34, DOI 10.1016/j.tpb.2009.03.007. (not covered in MR)
- Abraham Berman and Robert J. Plemmons, Nonnegative matrices in the mathematical sciences, Classics in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original. MR 1298430, DOI 10.1137/1.9781611971262
- Igor Chueshov, Monotone random systems theory and applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. MR 1902500, DOI 10.1007/b83277
- Miklós Farkas, Periodic motions, Applied Mathematical Sciences, vol. 104, Springer-Verlag, New York, 1994. MR 1299528, DOI 10.1007/978-1-4757-4211-4
- Barnabas M. Garay and Josef Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM J. Math. Anal. 34 (2003), no. 5, 1007–1039. MR 2001657, DOI 10.1137/S0036141001392815
- Russell A. Johnson, Kenneth J. Palmer, and George R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal. 18 (1987), no. 1, 1–33. MR 871817, DOI 10.1137/0518001
- Krešimir Josić and Robert Rosenbaum, Unstable solutions of nonautonomous linear differential equations, SIAM Rev. 50 (2008), no. 3, 570–584. MR 2429450, DOI 10.1137/060677057
- L. Yu. Kolotilina, Lower bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 180 (1993), 133–151. MR 1206413, DOI 10.1016/0024-3795(93)90528-V
- Tufail Malik and Hal L. Smith, Does dormancy increase fitness of bacterial populations in time-varying environments?, Bull. Math. Biol. 70 (2008), no. 4, 1140–1162. MR 2391183, DOI 10.1007/s11538-008-9294-5
- Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, Inc., New York, 1992. Reprint of the 1969 edition. MR 1215484
- Janusz Mierczyński, A simple proof of monotonicity for linear cooperative systems of ODEs, available at arXiv:1304.6562.
- Janusz Mierczyński and Sebastian J. Schreiber, Kolmogorov vector fields with robustly permanent subsystems, J. Math. Anal. Appl. 267 (2002), no. 1, 329–337. MR 1886831, DOI 10.1006/jmaa.2001.7776
- Janusz Mierczyński and Wenxian Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional systems, J. Math. Anal. Appl. 404 (2013), no. 2, 438–458. MR 3045185, DOI 10.1016/j.jmaa.2013.03.039
- Janusz Mierczyński and Wenxian Shen, Persistence in forward nonautonomous competitive systems of parabolic equations, J. Dynam. Differential Equations 23 (2011), no. 3, 551–571. MR 2836650, DOI 10.1007/s10884-010-9181-2
- Janusz Mierczyński and Wenxian Shen, Spectral theory for forward nonautonomous parabolic equations and applications, Infinite dimensional dynamical systems, Fields Inst. Commun., vol. 64, Springer, New York, 2013, pp. 57–99. MR 2986931, DOI 10.1007/978-1-4614-4523-4_{2}
- Janusz Mierczyński, Wenxian Shen, and Xiao-Qiang Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems, J. Differential Equations 204 (2004), no. 2, 471–510. MR 2085544, DOI 10.1016/j.jde.2004.02.014
- Henryk Minc, Nonnegative matrices, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 932967
- Paul L. Salceanu, Robust uniform persistence in discrete and continuous nonautonomous systems, J. Math. Anal. Appl. 398 (2013), no. 2, 487–500. MR 2990074, DOI 10.1016/j.jmaa.2012.09.005
- Sebastian J. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations 162 (2000), no. 2, 400–426. MR 1751711, DOI 10.1006/jdeq.1999.3719
- Allen J. Schwenk, Tight bounds on the spectral radius of asymmetric nonnegative matrices, Linear Algebra Appl. 75 (1986), 257–265. MR 825411, DOI 10.1016/0024-3795(86)90193-X
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438
Additional Information
- Janusz Mierczyński
- Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland
- Email: mierczyn@pwr.edu.pl
- Received by editor(s): May 25, 2013
- Published electronically: November 24, 2014
- Additional Notes: The author was supported by project S20058/I-18.
- Communicated by: Yingfei Yi
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1127-1135
- MSC (2010): Primary 34C12, 34D08, 37C65; Secondary 15B48, 92D25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12368-9
- MathSciNet review: 3293728