Invariant graph and random bony attractors
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In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.
Abstract
In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.
참고문헌 (35)
- Ya. I. Alber, "New results in operator theory and its applications", pp.7~22 (1997), Birkhauser
- L. Arnold, "Random dynamical systems, Springer Monographs in Mathematics" (1998), Springer-Verlag
- L. Arnold, "Diffusion processes and related problems in analysis Vol. II (Charlotte, NC, 1990)", pp.283~305 (1992), Birkhauser Boston
- A. Bielecki, "Iterated function system analogues on compact metric spaces and their attractors", Univ. Iagel. Acta Math, 32, pp.187~192 (1995)
- V. I. Bogachev, "Measure Theory Vol. I, II" (2007), Springer-Verlag
- D. Broomhead, "Random and deterministic perturbation of a class of skew-product systems", Dynam. Stability Systems, 14(2), pp.115~128 (1999)[DOI]
- H. Bruin, "Notes on ergodic theory", Preprint, ~ (2014)
- K. M. Campbell, "Observational noise in skew product systems", Phys. D, 107(1), pp.43~56 (1997)[DOI]
- K. M. Campbell, "The existence of inertial functions in skew product systems", Nonlinearity, 9(3), pp.801~817 (1996)[DOI]
- M. E. Davies, "Linear recursive filters and nonlinear dynamics", Nonlinearity, 9(2), pp.487~499 (1996)[DOI]
- L. J. Dıaz, "and phase transitions", Fund. Math, 216(1), pp.55~100 (2012)[DOI]
- J. H. Elton, "An ergodic theorem for iterated maps", Ergodic Theory Dynam. Systems, 7(4), pp.481~488 (1987)[DOI]
- J. H. Elton, "A multiplicative ergodic theorem for Lipschitz maps", Stochastic Process. Appl, 34(1), pp.39~47 (1990)[DOI]
- S. Fadaei, "Invariant graphs for chaotically driven maps", Nonlinearity, 31(11), pp.5329~5349 (2018)[DOI]
- H. Furstenberg, "Products of random matrices", Ann. Math. Statist, 31, pp.457~469 (1960)[DOI]
- M. W. Hirsch, "Stable manifolds for hyperbolic sets", Bull. Amer. Math. Soc, 75, pp.149~152 (1969)[DOI]
- M. W. Hirsch, "Invariant manifolds, Lecture Notes in Mathematics Vol. 583" (1977), Springer-Verlag
- B. R. Hunt, "Fractal dimensions of chaotic saddles of dynamical systems", Phys. Rev. E, 54, pp.4819~4823 (1996)
- B. R. Hunt, "Differentiable generalized synchronization of chaos", Phys. Rev. E, 55(4), pp.4029~4034 (1997)[DOI]
- J. Jachymski, "An iff fixed point criterion for continuous self-mappings on a complete metric space", Aequationes Math, 48(2-3), pp.163~170 (1994)[DOI]
- T. H. J¨ager, "Quasiperiodically forced interval maps with negative Schwarzian derivative", Nonlinearity, 16(4), pp.1239~1255 (2003)[DOI]
- T. J¨ager, "Random minimality and continuity of invariant graphs in random dynamical systems", Trans. Amer. Math. Soc, 368(9), pp.6643~6662 (2016)[DOI]
- V. Kleptsyn, "Physical measures for nonlinear random walks on interval", Mosc. Math. J, 14(2), pp.428~ (2014)[DOI]
- A. S. Kravchenko, "Completeness of the spaces of separable measures in the Kantrovich-Rubinshte˘ın metric", Siberian Math. J, 47(1), pp.85~96 (2006)[DOI]
- Yu. G. Kudryashov, "Bony attractors", Funct. Anal. Appl, 44(3), pp.219~222 (2010)
- L. M. Pecora, "Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data", Chaos, 6(3), pp.432~439 (1996)[DOI]
- M. Rabiee, "An open set of skew products with invariant multi-graphs and bony multi-graphs", Qual. Theory Dyn. Syst, 21(4) (2022)[DOI]
- J. Stark, "Regularity of invariant graphs for forced systems", Ergodic Theory Dynam. Systems, 19(1), pp.155~199 (1999)[DOI]
- J. Stark, "Recursive filters driven by chaotic signals", IEEE Colloquium on Exploiting Chaos in Signal Processing, 1, pp.431~516 (1994)
- O. Stenflo, "A survey of average contractive iterated function systems", J. Difference Equ. Appl, 18(8), pp.1355~1380 (2012)[DOI]
- R. Temam, "Infinite-dimensional dynamical systems in mechanics and physics" (1988), Springer-Verlag
- M. Viana, "Foundations of ergodic theory" (2016), Cambridge University Press
- P. Walters, "Introduction to Ergodic Theory" (1982), Springer-Verlag
- M. Zaj, "Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus", Discrete Contin. Dyn. Syst, 38(4), pp.1777~1807 (2018)[DOI]
- M. Zaj, "Non hyperbolic solenoidal thick bony attractors", Qual. Theory Dyn. Syst, 18(1), pp.35~55 (2019)[DOI]