On the global attractivity of monotone random dynamical systems

Cao, Feng; Jiang, Jifa

doi:10.1090/S0002-9939-09-09912-2

On the global attractivity of monotone random dynamical systems
HTML articles powered by AMS MathViewer

by Feng Cao and Jifa Jiang PDF

Proc. Amer. Math. Soc. 138 (2010), 891-898 Request permission

Abstract:

Suppose that $(\theta ,\varphi )$ is a monotone (order-preserving) random dynamical system (RDS for short) with state space $V$, where $V$ is a real separable Banach space with a normal solid minihedral cone $V_{+}$. It is proved that the unique equilibrium of $(\theta ,\varphi )$ is globally attractive if every pull-back trajectory has compact closure in $V$.

References

Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Igor Chueshov, Monotone random systems theory and applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. MR 1902500, DOI 10.1007/b83277
E. N. Dancer and P. Poláčik, Realization of vector fields and dynamics of spatially homogeneous parabolic equations, Mem. Amer. Math. Soc. 140 (1999), no. 668, viii+82. MR 1618487, DOI 10.1090/memo/0668
E. N. Dancer, Some remarks on a boundedness assumption for monotone dynamical systems, Proc. Amer. Math. Soc. 126 (1998), no. 3, 801–807. MR 1443378, DOI 10.1090/S0002-9939-98-04276-2
G. A. Enciso and E. D. Sontag, Global attractivity, I/O monotone small-gain theorems, and biological delay systems, Discrete Contin. Dyn. Syst. 14 (2006), no. 3, 549–578. MR 2171727, DOI 10.3934/dcds.2006.14.549
Georg Hetzer, Wenxian Shen, and Shu Zhu, Asymptotic behavior of positive solutions of random and stochastic parabolic equations of Fisher and Kolmogorov types, J. Dynam. Differential Equations 14 (2002), no. 1, 139–188. MR 1878647, DOI 10.1023/A:1012932212645
Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1
Ji Fa Jiang, On the global stability of cooperative systems, Bull. London Math. Soc. 26 (1994), no. 5, 455–458. MR 1308362, DOI 10.1112/blms/26.5.455
Ji-Fa Jiang, On the analytic order-preserving discrete-time dynamical systems in $\mathbf R^n$ with every fixed point stable, J. London Math. Soc. (2) 53 (1996), no. 2, 317–324. MR 1373063, DOI 10.1112/jlms/53.2.317
J. F. Jiang, Sublinear discrete-time order-preserving dynamical systems, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 561–574. MR 1357065, DOI 10.1017/S0305004100074417
Ji-Fa Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal. 27 (1996), no. 6, 1738–1744. MR 1416516, DOI 10.1137/S003614109326063X
Ji Fa Jiang, A note on a global stability theorem of M. W. Hirsch, Proc. Amer. Math. Soc. 112 (1991), no. 3, 803–806. MR 1043411, DOI 10.1090/S0002-9939-1991-1043411-0
Ji-Fa Jiang and Shu-Xiang Yu, Stable cycles for attractors of strongly monotone discrete-time dynamical systems, J. Math. Anal. Appl. 202 (1996), no. 1, 349–362. MR 1402605, DOI 10.1006/jmaa.1996.0320
Jifa Jiang and Xiao-Qiang Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math. 589 (2005), 21–55. MR 2194677, DOI 10.1515/crll.2005.2005.589.21
Patrick De Leenheer, David Angeli, and Eduardo D. Sontag, Monotone chemical reaction networks, J. Math. Chem. 41 (2007), no. 3, 295–314. MR 2343862, DOI 10.1007/s10910-006-9075-z
Patrick De Leenheer, Simon A. Levin, Eduardo D. Sontag, and Christopher A. Klausmeier, Global stability in a chemostat with multiple nutrients, J. Math. Biol. 52 (2006), no. 4, 419–438. MR 2235513, DOI 10.1007/s00285-005-0344-4
Sylvia Novo, Rafael Obaya, and Ana M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations 235 (2007), no. 2, 623–646. MR 2317498, DOI 10.1016/j.jde.2006.12.009
Peter Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), no. 1, 89–110. MR 997611, DOI 10.1016/0022-0396(89)90115-0
P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 339–360. MR 1132766, DOI 10.1007/BF00375672
P. Poláčik and Ignác Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Differential Equations 5 (1993), no. 2, 279–303. MR 1223450, DOI 10.1007/BF01053163
S. Smale, On the differential equations of species in competition, J. Math. Biol. 3 (1976), no. 1, 5–7. MR 406579, DOI 10.1007/BF00307854
H. L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equations Appl. 3 (1998), no. 5-6, 335–357. MR 1618123, DOI 10.1080/10236199708808108
Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
Hal L. Smith and Horst R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal. 22 (1991), no. 4, 1081–1101. MR 1112067, DOI 10.1137/0522070
Eduardo D. Sontag, Molecular systems biology and control, Eur. J. Control 11 (2005), no. 4-5, 396–435. MR 2201569, DOI 10.3166/ejc.11.396-435