PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 1 | 1 |
Tytuł artykułu

Estimates for Principal Lyapunov Exponents: A Survey

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Rocznik
Tom
1
Numer
1
Opis fizyczny
Daty
otrzymano
2014-03-17
zaakceptowano
2014-09-01
online
2014-11-12
Twórcy
  • Institute of Mathematics and Computer Science, Wrocław University of Technology,
    Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland, mierczyn@pwr.edu.pl
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, ClassicsAppl. 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 numericalresults, 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 bySpringer. 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. MR1625845 (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 Floquetbundles 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. MR1206413 (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, NewYork, 1992. MR 0162808 (29 #112)
  • [27] J. Mierczyński, Globally positive solutions of linear parabolic partial differential equations of second order with Dirichletboundary 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, ChapmanHall/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 DimensionalDynamical 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, NagoyaMath. 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 dispersaloperators 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. DifferentialEquations 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. Availableat 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)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_msds-2014-0008
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.