Semiattractors of set-valued semiflows. (English) Zbl 1398.37016
Summary: We consider smallest closed invariant subsets of the phase space with respect to arbitrary set-valued semiflows. Such sets, called semiattractors, attract all trajectories of singletons but not necessary compact or bounded subsets. In particular, there are systems admitting semiattractors which do not have global attractors as usually understood. We show some sufficient conditions on the existence of such a set. We also prove lower semicontinuous dependence of semiattractors for strict semiflows of l.s.c. multifunction and the existence of semigroups of Markov operators generated by such semiflows. We are motivated by asymptotic properties of semiflows of multifunctions connected with iterated function systems as well as random dynamical systems.
MSC:
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
Keywords:
semiattractor; semifractal; set-valued semiflow; iterated function system; random dynamical system; semigroup of Markov operatorsReferences:
| [1] | Arnold, L., Random Dynamical Systems (1998), Springer: Springer Berlin · Zbl 0906.34001 |
| [2] | Balibrea, F.; Caraballo, T.; Kloeden, P. E.; Valero, J., Recent developments in dynamical systems: three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20, 9, 2591-2636 (2010) · Zbl 1202.37003 |
| [3] | Barnsley, M. F., Fractals Everywhere (1988), Academic Press: Academic Press New York · Zbl 0691.58001 |
| [4] | Burachik, R. S.; Iusem, A. N., Set-Valued Mappings and Enlargements of Monotone Operators (2008), Springer: Springer New York · Zbl 1154.49001 |
| [5] | Cheban, D. N., Global Attractors of Nonautonomous Dissipative Dynamical Systems (2004), World Scientific: World Scientific River Edge, New Jersey · Zbl 1098.37002 |
| [6] | Cheban, D. N., Compact global attractors of control systems, J. Dyn. Control Syst., 16, 1, 23-44 (2010) · Zbl 1203.37027 |
| [7] | Cheban, D. N., Compact global chaotic attractors of discrete control systems, Nonaut. Stoch. Dyn. Syst., 1, 10-25 (2014) · Zbl 1338.37023 |
| [8] | Cheban, D. N.; Mammana, C., Continuous dependence of attractors on parameters of nonautonomous dynamical systems and infinite iterated function systems, Discrete Contin. Dyn. Syst., 18, 2-3, 499-515 (2007) · Zbl 1138.37006 |
| [9] | Chueshov, I., Monotone Random Systems Theory and Applications, Lecture Notes in Math., vol. 1779 (2002), Springer · Zbl 1023.37030 |
| [10] | Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probab. Theory Related Fields, 100, 365-393 (1994) · Zbl 0819.58023 |
| [11] | Crauel, H., Random point attractors versus random set attractors, J. Lond. Math. Soc. (2), 63, 413-427 (2001) · Zbl 1011.37032 |
| [12] | Crauel, H., Measure attractors and Markov attractors, Dyn. Syst., 23, 75-107 (2008) · Zbl 1153.37391 |
| [13] | de Freitas, M. M.; Sontag, E. D., Random dynamical systems with inputs, (Kloeden, P. E.; Potzsche, Ch., Nonautonomous Dynamical Systems in the Life Sciences. Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Math., vol. 2102 (2013), Springer), 41-87 · Zbl 1311.37038 |
| [14] | Guzik, G., Asymptotic stability of discrete cocycles, J. Difference Equ. Appl., 21, 11, 1044-1057 (2015) · Zbl 1337.28013 |
| [15] | Guzik, G., Cocycles and continuous iteration semigroups of triangular functions, J. Difference Equ. Appl. (2015), in press · Zbl 1341.39008 |
| [16] | Guzik, G., Asymptotic properties of multifunctions, families of measures and Markov operators associated with cocycles, Nonlinear Anal., 130, 59-75 (2016) · Zbl 1360.37136 |
| [17] | Hoang, L. T.; Olson, E. J.; Robinson, J. C., On continuity of global attractors, Proc. Amer. Math. Soc. (2015), in press · Zbl 1356.37027 |
| [18] | Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30, 713-747 (1981) · Zbl 0598.28011 |
| [19] | Jachymski, J., Continuous dependence of attractors of iterated function system, J. Math. Anal. Appl., 198, 221-226 (1996) · Zbl 0862.54033 |
| [20] | Kloeden, P. E.; Rassmusen, M., Nonautonomous Dynamical Systems, Math. Surveys Monogr., vol. 176, 28-37 (2011), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 1244.37001 |
| [21] | Kuratowski, K., Topology, vol. I (1966), Academic Press: Academic Press New York · Zbl 0158.40901 |
| [22] | Kwiecińska, G., A note on the set of discontinuous points of multifunctions of two variables, Tatra Mt. Math. Publ., 58, 145-154 (2014) · Zbl 1413.54073 |
| [23] | Lasota, A.; Myjak, J., Semifractals, Bull. Pol. Acad. Sci. Math., 44, 1, 5-21 (1996) · Zbl 0847.28006 |
| [24] | Lasota, A.; Myjak, J., Semifractals on Polish spaces, Bull. Pol. Acad. Sci. Math., 46, 2, 179-196 (1998) · Zbl 0921.28007 |
| [25] | Lasota, A.; Myjak, J., Attractors of multifunctions, Bull. Pol. Acad. Sci. Math., 48, 3, 319-334 (2000) · Zbl 0962.28004 |
| [26] | Lasota, A.; Myjak, J.; Szarek, T., Markov operators with a unique invariant measure, J. Math. Anal. Appl., 276, 343-356 (2002) · Zbl 1009.60059 |
| [27] | Melnik, V. S.; Valero, J., On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 6, 83-111 (1998) · Zbl 0915.58063 |
| [28] | Michael, E., Continuous selections I, Ann. of Math., 63, 361-381 (1956) · Zbl 0071.15902 |
| [29] | Szarek, T.; Śleczka, M.; Urbański, M., On stability of velocity vectors for some passive tracer models, Bull. Lond. Math. Soc., 42, 923-936 (2010) · Zbl 1204.60070 |
| [30] | Worm, D. T.H.; Hille, S. C., An ergodic decomposition defined by regular jointly measurable Markov semigroups on Polish spaces, Acta Appl. Math., 116, 27-53 (2011) · Zbl 1252.60073 |
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