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Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional systems. (English) Zbl 1320.37027
The authors’ results from [Trans. Am. Math. Soc. 365, No. 10, 5329–5365 (2013; Zbl 1350.37061)], on what the authors call principal Lyapunov exponents and principal Floquet subspaces, are specialized for positive linear random dynamical systems on finite-dimensional ordered normed vector spaces. Conditions are given for systems generated by products of non-negative matrices and by strongly co-operative or by type-\(K\) strongly monotone linear differential equations to admit measurable families of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential splittings.

MSC:
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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[1] Abramovich, Y. A.; Aliprantis, C. D.; Burkinshaw, O., Positive operators on Kreĭn spaces, Acta Appl. Math., 27, 1-2, 1-22, (1992), MR 93h:47045 · Zbl 0785.47029
[2] Arnold, L., (Random Dynamical Systems, Springer Monogr. Math., (1998), Springer Berlin), MR 2000m:37087
[3] Arnold, L.; Gundlach, V. M.; Demetrius, L., Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., 4, 3, 859-901, (1994), MR 95h:28028 · Zbl 0818.15015
[4] Benaïm, M.; Schreiber, S. J., Persistence of structured populations in environmental models, Theor. Popul. Biol., 76, 1, 19-34, (2009) · Zbl 1213.92057
[5] Chueshov, I., (Monotone Random Systems Theory and Applications, Lecture Notes in Math., vol. 1779, (2002), Springer Berlin), MR 2003d:37072 · Zbl 1023.37030
[6] Gyllenberg, M.; Wang, Y., Dynamics of the periodic type-\(K\) competitive Kolmogorov systems, J. Differential Equations, 205, 1, 50-76, (2004), MR 2005f:34129 · Zbl 1064.34031
[7] Hirsch, M. W., Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383, 1-53, (1988), MR 89c:58108 · Zbl 0624.58017
[8] Hirsch, M. W.; Smith, H. L., Monotone dynamical systems, (Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, (2005), Elsevier Amsterdam), 239-357, MR 2006j:37017 · Zbl 1094.34003
[9] Johnson, R.; Palmer, K.; Sell, G. R., Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18, 1, 1-33, (1987), MR 88a:58112 · Zbl 0641.58034
[10] Kon, R.; Iwasa, Y., Single-class orbits in nonlinear Leslie matrix models for semelparous populations, J. Math. Biol., 55, 5-6, 781-802, (2007), MR 2008h:92069 · Zbl 1145.37043
[11] Krengel, U., Ergodic theorems, (1985), Walter de Gruyter Berlin, MR 87i:28001 · Zbl 0471.28011
[12] Lakshmikantham, V.; Leela, S., (Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations, Math. Sci. Eng., vol. 55-I, (1969), Academic Press New York, London), MR 52 #837 · Zbl 0177.12403
[13] Lian, Z.; Lu, K., Lyapunov exponents and invariant manifolds for random dynamical systems on a Banach space, Mem. Amer. Math. Soc., 206, 967, (2010), MR 2011g:37145
[14] Liang, X.; Jiang, J., The dynamical behaviour of type-\(K\) competitive Kolmogorov systems and its application to three-dimensional type-\(K\) competitive Lotka-Volterra systems, Nonlinearity, 16, 3, 785-801, (2003), MR 2004m:37167 · Zbl 1042.34067
[15] Mañé, R., Ergodic theory and differentiable dynamics, (Ergeb. Math. Grenzgeb. (3), (1987), Springer Berlin), translated from the Portuguese by S. Levy. MR 88c:58040
[16] Mierczyński, J.; Shen, W., (Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 139, (2008), CRC Press Boca Raton, FL), MR 2010g:35216 · Zbl 1387.35007
[17] J. Mierczyński, W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory, Trans. Amer. Math. Soc., in press (http://dx.doi.org/10.1090/S0002-9947-2013-05814-X). Preprint available at http://arxiv.org/abs/1209.3475. · Zbl 1350.37061
[18] Millionshchikov, V. M., Metric theory of linear systems of differential equations, Math. USSR-Sb., 6, 149-158, (1968), MR 38 #383
[19] Oseledets, V. I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow. Math. Soc., 19, 197-231, (1968), MR 39 #1629 · Zbl 0236.93034
[20] Raghunathan, M. S., A proof of oseledec’s multiplicative ergodic theorem, Israel J. Math., 32, 4, 356-362, (1979), MR 81f:60016 · Zbl 0415.28013
[21] Ruelle, D., Ergodic theory of differentiable dynamical systems, Inst. Hautes Ètudes Sci. Publ. Math. No., 50, 27-58, (1979), MR 81f:58031 · Zbl 0426.58014
[22] Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115, 2, 243-290, (1982), MR 83j:58097 · Zbl 0493.58015
[23] Schaefer, H. H., (Topological Vector Spaces, Grad. Texts in Math., vol. 3, (1980), Springer New York, Berlin), fourth printing corrected. MR 49 #7722 · Zbl 0435.46003
[24] Schaumlöffel, K.-U.; Flandoli, F., A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain, Stoch. Stoch. Rep., 34, 3-4, 241-255, (1991), MR 92m:60050 · Zbl 0724.60072
[25] Smith, H. L., Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64, 2, 165-194, (1986), MR 87k:92027 · Zbl 0596.34013
[26] Smith, H. L., (Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41, (1995), American Mathematical Society Providence, RI), MR 96c:34002 · Zbl 0821.34003
[27] Walter, W., On strongly monotone flows, Ann. Polon. Math., 66, 269-274, (1997), MR 98b:34067 · Zbl 0870.34014
[28] Wang, Y.; Jiang, J., The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal. RWA, 3, 4, 471-485, (2002), MR 2003g:34112 · Zbl 1020.92026
[29] Zhao, X.-Q., Global asymptotic behavior in a periodic competitor-competitor-mutualist parabolic system, Nonlinear Anal., 29, 5, 551-568, (1997), MR 99d:92034 · Zbl 0876.35058
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