Estimates for Principal Lyapunov Exponents: A Survey

• Autorzy: Janusz Mierczyński
• Języki publikacji: EN

Abstrakty

EN

This is a survey of known results on estimating the principal Lyapunov exponent of a timedependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further research are hinted.

Słowa kluczowe

EN

Principal Lyapunov exponent; principal spectrum; nonautonomous dynamical system; randomdynamical system; upper and lower estimate; time-averaging; strongly cooperative system of ordinary differentialequations; dominant eigenvalue; parabolic partial differential equation; principal eigenvalue; permanence

• Wydawca: De Gruyter Open
• Czasopismo: Nonautonomous Dynamical Systems
• Rocznik: 2014
• Tom: 1
• Numer: 1

Daty

otrzymano

2014-03-17

zaakceptowano

2014-09-01

online

2014-11-12

Twórcy

autor

Janusz Mierczyński

• Institute of Mathematics and Computer Science, Wrocław University of Technology,
  Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland

Bibliografia

• [1] L. Arnold, Random dynamical systems, Springer Monogr. Math. Springer, Berlin, 1998. MR 1723992 (2000m:37087)
• [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)
• [3] J. Banasiak, L. Arlotti, Perturbations of positive semigroups with applications, Springer Monogr. Math. Springer, London, 2006. MR 2178970 (2006i:47076)
• [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)
• [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.
• [7] C. Chicone, Ordinary differential equations with application, second edition, Texts Appl. Math., 34. Springer, New York, 2006. MR 2224508 (2006m: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)
• [9] R. Courant, D. Hilbert, Methods of mathematical physics, Vol. I. Interscience Publishers, New York, 1953. MR 0065391 (16,426a)
• [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)
• [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)
• [13] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19. American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
• [14] M. Farkas, Periodic motions, Appl. Math. Sci., 104. Springer, New York, 1994. MR 1299528 (95g:34058)
• [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)
• [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) [WoS]
• [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)
• [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) [WoS]
• [22] A. Iserles, Expansions that grow on trees, Notices Amer.Math. Soc. 49 (2002), no. 4, 430–440. MR 1892640 (2003b:34022)
• [23] K. Josić, R. Rosenbaum, Unstable solutions of nonautonomous linear differential equations, SIAM Rev. 50 (2008), no. 3, 570–584. MR 2429450 (2009d:34128)
• [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)
• [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) [WoS]
• [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)
• [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.
• [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)
• [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)
• [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
• [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 [WoS]
• [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 [WoS]
• [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)
• [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. [WoS][Crossref]
• [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 [WoS]
• [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 [WoS][Crossref]
• [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)
• [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)
• [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)
• [45] S. Tuljapurkar, Population dynamics in variable environments, Lect. Notes in Biomath., 85. Springer, Berlin–Heidelberg, 1990. (not covered in MR)
• [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)

Identyfikatory

DOI

10.2478/msds-2014-0008