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Estimates for principal Lyapunov exponents: a survey. (English) Zbl 1316.34004
This survey is devoted to the estimate of principal Lyapunov exponents in time-dependent linear systems of differential equations and some of their generalizations.
One of the characteristic results is as follows. Consider a system $$T$$-periodic in $$t$$ $u'=P(t)u$ and assume that the system is strongly cooperative (i.e., the off-diagonal entries of the matrix $$P(t)$$ are positive for all $$t$$). Then there exist (and are uniquely determined) a number $$\rho>0$$ and a positive solution $$v(t)$$ with $$|v(0)|=1$$ such that $$v(T)=\rho v(0)$$. In this case, $\lim_{t\to\infty}(\log|u(t)|)/t=(\log\rho)/T$ for any positive solution $$u(t)$$.
The survey contains a lot of results for second order parabolic PDEs, almost periodic linear systems of differential equations, systems with random coefficients, cooperative delay differential equations, and so on.
Concerning proofs, mostly hints are given.

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34D08 Characteristic and Lyapunov exponents of ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs 37C65 Monotone flows as dynamical systems 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents 34A30 Linear ordinary differential equations and systems 37C60 Nonautonomous smooth dynamical systems 34K06 Linear functional-differential equations
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##### References:
 [1] L. Arnold, Random dynamical systems, Springer Monogr. Math. Springer, Berlin, 1998. MR 1723992 (2000m:37087) · Zbl 0906.34001 [2] L. Arnold, V. M. Gundlach, L. Demetrius, Evolutionary formalism for products of positive randommatrices, Ann. Appl. Probab. 4 (1994), no. 3, 859-901. MR 1284989 (95h:28028) · Zbl 0818.15015 [3] J. Banasiak, L. Arlotti, Perturbations of positive semigroups with applications, Springer Monogr. Math. Springer, London, 2006. MR 2178970 (2006i:47076) · Zbl 1097.47038 [4] M. Benaïm, S. J. Schreiber, Persistence of structured populations in random environments, Theor. Popul. Biol. 76 (2009), no. 1, 19-34. (not covered in MR) · Zbl 1213.92057 [5] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, revised reprint of the 1979 original, Classics Appl. Math., 9. SIAM, Philadelphia, PA, 1994. MR 1298430 (95e:15013) [6] J. A. Calzada, R. Obaya, A. M. Sanz, Continuous separation for monotone skew-product semiflows: From theoretical to numerical results, Discrete Contin. Dyn. Syst. Ser. B, in press. · Zbl 1366.37061 [7] C. Chicone, Ordinary differential equations with application, second edition, Texts Appl. Math., 34. Springer, New York, 2006. MR 2224508 (2006m:34001) · Zbl 1120.34001 [8] C. Chicone, Y. Latushkin, Evolution semigroups in dynamical systems and differential equations,Math. Surveys Monogr., 70. American Mathematical Society, Providence, RI, 1999. MR 1707332 (2001e:47068) · Zbl 0970.47027 [9] R. Courant, D. Hilbert, Methods of mathematical physics, Vol. I. Interscience Publishers, New York, 1953. MR 0065391 (16,426a) · Zbl 0053.02805 [10] R. Dautray, J.-L. Lions,Mathematical analysis and numerical methods for science and technology. Vol. 5, evolution problems. I, with the collaboration of M. Artola, M. Cessenat and H. Lanchon, translated from the French by A. Craig. Springer, Berlin, 1992. MR 1156075 (92k:00006) · Zbl 0755.35001 [11] K. Deimling, Nonlinear functional analysis, unabridged, emended republication of the 1985 edition originally published by Springer. Dover Publications, New York, 2010. MR 0787404 (86j:47001) [12] K.-J. Engel, R. Nagel, A short course on operator semigroups, Universitext. Springer, New York, 2006. MR 2229872 (2007e:47001) · Zbl 1106.47001 [13] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19. American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001) · Zbl 0902.35002 [14] M. Farkas, Periodic motions, Appl. Math. Sci., 104. Springer, New York, 1994. MR 1299528 (95g:34058) · Zbl 0805.34037 [15] A. M. Fink, Almost periodic differential equations, Lect. Notes in Math., 377. Springer, Berlin-New York, 1974. MR 0460799 (57 #792) [16] A. Friedman, Partial differential equations of parabolic type, unabridged republication of the 1964 edition. Dover Publications, New York, 2008. MR 0181836 (31 #6062) [17] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes Math. Ser., 247. Longman/Wiley, Harlow/New York, 1991. MR 1100011 (92h:35001) · Zbl 0731.35050 [18] M.W. Hirsch, H. L. Smith, Monotone dynamical systems. Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, 239-357. Elsevier, Amsterdam, 2005. MR 2182759 (2006j:37017) [19] J. Húska, P. Poláčik, M. V. Safonov, Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 5, 711-739. MR 2348049 (2008k:35211) · Zbl 1139.35046 [20] V. Hutson,W. Shen, G. T. Vickers, Estimates for the principal spectrumpoint for certain time-dependent parabolic operators, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1669-1679. MR 1814096 (2001m:35243) · Zbl 0963.35074 [21] V. Hutson, W. Shen, G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math. 38 (2008), no. 4, 1147-1175. MR 2436718 (2009g:47197) · Zbl 1255.47075 [22] A. Iserles, Expansions that grow on trees, Notices Amer.Math. Soc. 49 (2002), no. 4, 430-440. MR 1892640 (2003b:34022) · Zbl 1126.34310 [23] K. Josić, R. Rosenbaum, Unstable solutions of nonautonomous linear differential equations, SIAM Rev. 50 (2008), no. 3, 570-584. MR 2429450 (2009d:34128) · Zbl 1156.34008 [24] L. Y. Kolotilina, Lower bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 180 (1993), 133-151. MR 1206413 (94b:15016) · Zbl 0785.15005 [25] T. Malik, H. L. Smith, Does dormancy increase fitness of bacterial populations in time-varying environments?, Bull. Math. Biol. 70 (2008), no. 4, 1140-1162. MR 2391183 (2009g:92091) · Zbl 1142.92045 [26] M. Marcus, H. Minc, A survey of matrix theory and matrix inequalities, reprint of the 1969 edition. Dover Publications, New York, 1992. MR 0162808 (29 #112) [27] J. Mierczyński, Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions, J. Math. Anal. Appl. 226 (1998), no. 2, 326-347. MR 1650236 (99m:35096) · Zbl 0921.35064 [28] J. Mierczyński, Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs, Proc. Amer.Math. Soc., in press. Available at arXiv:1305.6198. · Zbl 1311.34123 [29] J. Mierczyński,W. Shen, The Faber-Krahn inequality for random/nonautonomous parabolic equations, Commun. Pure Appl. Anal. 4 (2005), no. 1, 101-114. MR 2126280 (2006b:35358) · Zbl 1194.35520 [30] J. Mierczyński, W. Shen, Spectral theory for random and nonautonomous parabolic equations and applications, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL, 2008. MR 2464792 (2010g:35216) · Zbl 1387.35007 [31] J. Mierczyński, W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications. Infinite Dimensional Dynamical Systems, 57-99, Fields Inst. Commun., 64. Springer, New York, 2013. MR 2986931 · Zbl 1274.35253 [32] J. Mierczyński,W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5329-5365. MR 3074376 · Zbl 1350.37061 [33] J. Mierczyński,W. 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 · Zbl 1320.37027 [34] M. Nishio, The uniqueness of positive solutions of parabolic equations of divergence form on an unboundeddomain, Nagoya Math. J. 130 (1993), 111-121. MR 1223732 (94f:35058) · Zbl 0774.31006 [35] S. Novo, R. Obaya, A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynam. Differential Equations 25 (2013), no. 4, 1201-1231. · Zbl 1291.54045 [36] V. Y. Protasov, R. M. Jungers, Lower and upper bounds for the largest Lyapunov exponent of matrices, Linear Algebra Appl. 438 (2013), no. 11, 4448-4468. MR 3034543 · Zbl 1281.65154 [37] M. H. Protter, H. F. Weinberger,Maximum principles in differential equations, corrected reprint of the 1967 original. Springer, New York, 1984. MR 0762825 (86f:35034) [38] N. Rawal, W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations 24 (2012), no. 4, 927-954. MR 3000610 · Zbl 1258.35155 [39] J. B. T. M. Roerdink, The biennial life strategy in a random environment, J.Math. Biol. 26 (1988), no. 2, 199-215. MR 0946177 (90i:92030) · Zbl 0713.92028 [40] S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence, Proc. Roy. Soc. Edinburgh Sect. B 277 (2010), 1907-1914. (not covered in MR) [41] E. Seneta, Non-negative matrices and Markov chains, revised reprint of the second (1981) edition, Springer Ser. Statist. Springer, New York, 2006. MR 2209438 [42] W. Shen, G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations 235 (2007), no. 1, 262-297. MR 2309574 (2008d:35091) · Zbl 1117.35057 [43] W. Shen, X. Xie, On principal spectrumpoints/principal eigenvalues of nonlocal dispersal operators and applications. Available at arXiv:1309.4753. [44] W. Shen, Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), no. 647. MR 1445493 (99d:34088) · Zbl 0913.58051 [45] S. Tuljapurkar, Population dynamics in variable environments, Lect. Notes in Biomath., 85. Springer, Berlin-Heidelberg, 1990. (not covered in MR) · Zbl 0704.92014 [46] W.Walter, Differential and Integral Inequalities, translated from the German by L. Rosenblatt and L. Shampine, Ergeb.Math. Grenzgeb., 55. Springer, New York-Berlin, 1970. MR 0271508 (42 #6391) [47] A. Wintner, Asymptotic integration constants, Amer. J. Math. 68 (1946), no. 4, 553-559. MR 0018310 (8,272f) · Zbl 0063.08295
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