Topological Methods in Nonlinear Analysis

Weak forms of shadowing in topological dynamics

Danila Cherkashin and Sergey Kryzhevich

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Abstract

We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of $\varepsilon$-networks ($\varepsilon > 0$) whose iterations are also $\varepsilon$-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 1 (2017), 125-150.

Dates
First available in Project Euclid: 14 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1507946573

Digital Object Identifier
doi:10.12775/TMNA.2017.020

Mathematical Reviews number (MathSciNet)
MR3706154

Zentralblatt MATH identifier
06850993

Citation

Cherkashin, Danila; Kryzhevich, Sergey. Weak forms of shadowing in topological dynamics. Topol. Methods Nonlinear Anal. 50 (2017), no. 1, 125--150. doi:10.12775/TMNA.2017.020. https://projecteuclid.org/euclid.tmna/1507946573


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References

  • D. Angeli, J.E. Ferrell Jr. and E.D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci. USA 101 (2003), 1822–1827.
  • D.V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova 90 (1967), 209 (in Russian).
  • N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, vol. 52, North-Holland Math. Library, North-Holland, Amsterdam, 1994.
  • V.I. Babitsky, Theory of Vibro-Impact Systems and Applications. Springer, Berlin, 1998.
  • N. Bernardes and U. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math. 231 (2012), 1655–1680.
  • M. di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer, London, 2008.
  • S. Bezuglyi and S. Kolyada (eds.), Topics in Dynamics and Ergodic Theory, London Mathematical Society Lecture Note Series, Vol. 310, Cambridge University Press, Cambridge, 2003.
  • M.L. Blank, Metric properties of $\varepsilon$-trajectory of dynamical systems with stochastic behavior, Ergodic Theory Dynam. Systems 8, 1988, 365–378.
  • C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), 33–104 (in French).
  • C. Bonatti, L.G. Diaz, and G. Turcat, Pas de shadowing lemma pour des dynamiques partiellement hyperboliques, C. R. Math. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 587–592 (in French).
  • R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975.
  • I.U. Bronstein, Extensions of Minimal Transformation Groups, American Mathematical Society, Providence, 1988.
  • Ch. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, Vol. 38. American Mathematical Society, Providence, 1978.
  • D.A. Dastjerdi and M. Hosseini, Shadowing with chain transitivity, Topology Appl. 156 (2009), 2193–2195.
  • D.A. Dastjerdi and M. Hosseini, Sub-shadowings, Nonlinear Anal. 72 (2010), 3759–3766.
  • A. Fakhari and F.H. Gane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364 (2010), 151–155.
  • B. Feeny and F.C. Moon, Chaos in a forced dry-friction oscillator: experiments and numerical modelling, J. Sound Vib. 170 (1994), 303–323.
  • E. Glasner, Classifying dynamical systems by their recurrence properties, Methods Nonlinear Anal. 24 (2004), 21–40.
  • E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), 1067–1075.
  • W.H. Gottschalk and G.A. Hedlund, Topological dynamics, Bull. Amer. Math. Soc. 61 (1955), no. 6, 584–588.
  • P.R. Halmos, Measure Theory, Springer, New York, 1950.
  • R.A. Ibrahim, Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer, Berlin, 2009.
  • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1997.
  • J.F.C. Kingman, Subadditive ergodic theory, Ann. Probab. 1 (1973), 883–899.
  • P. Kościelnak, M. Mazur, P. Oprocha and P. Pilarczyk, Shadowing is generic–-a continuous map case, Discrete. Contin. Dyn. Syst. 34 (2014), 3591–3609.
  • S.G. Kryzhevich, Shadowing along subsequences for continuous mappings, Vestnik St. Petersburg Univ. Math. 47 (2014), no. 3, 102–104.
  • K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete and Contin. Dyn. Syst. 13 (2005), no. 2, 533–539.
  • J.H. Mai and X. Ye, The structure of pointwise recurrent maps having the pseudo-orbit tracing property, Nagoya Math. J. 166 (2002), 83–92.
  • O. Makarenkov and J.S.W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D: Nonlinear Phenomena 241 (2012), 1826–1844.
  • M. Mazur and P. Oprocha, S-limit shadowing is $C^0$-dense, J. Math. Anal. Syst. 408 (2013), 465–475.
  • T.K.S. Moothathu, Implications of pseudo-orbit tracing property for continuous maps on compacta, Topology Appl. 158 (2011), 2232–2239.
  • T.K.S. Moothathu and P. Oprocha, Shadowing, entropy and minimal subsystems, Montash Math. 172 (2013), 357–378.
  • V.V. Nemytskiĭ and V.V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Dover, New York, 1989.
  • A.V. Osipov, S.Yu. Pilyugin and S.B. Tikhomirov, Periodic shadowing and $\Omega$-stability, Regul. Chaotic Dyn. 15 (2010), 404–417.
  • K.J. Palmer, Shadowing in Dynamical Systems: Theory and Applications, Kluwer, Dordrecht, 2000.
  • K.J. Palmer, S.Yu. Pilyugin and S.B. Tikhomirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations 252 (2012), 1723–1747.
  • S.Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, Vol. 1706, Springer, Berlin, 1999.
  • ––––, The Space of Dynamical Systems with the $C^0$-Topology, Lecture Notes in Mathematics, Vol. 1571, Springer, Berlin, 1994.
  • S.Yu. Pilyugin and O.B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), 253–266.
  • S.Yu. Pilyugin and K. Sakai, $C^0$ transversality and shadowing properties, Tr. Mat Inst. Steklova 256 (2007), 290–305.
  • S.Yu. Piljugin and K. Sakai, Transversality and shadowing properties, Tr. Mat. Inst. Steklova 256 (2007), 305–319.
  • S.Yu. Pilyugin and S.B. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity 23 (2010), 2509–2515.
  • M. Reed and B. Simon, Methods of modern mathematical physics: Functional Analysis, Vol. I, Academic Press, New York, 1973.
  • C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), 425–437.
  • K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds, Osaka J. Math, 31 (1994), 373–386.
  • ––––, Shadowing Property and Transversality Condition, Dynamical Systems and Chaos, Vol. 1, World Scientific, River Edge, 1995, 233–238.
  • ––––, Various shadowing properties for positively expansive maps, Topology Appl. 131 (2003), 15–31.
  • K. Sawada, Extended $f$-orbits are approximated by orbits, Nagoya Math. J. 79 (1980), 33–45.
  • S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.
  • J. de Vries, Topological Dynamical Systems: An Introduction to the Dynamics of Continuous Mappings, De Gruyter Studies in Mathematics Vol. 59, De Gruyter, 2014.
  • G.-C. Yuan and J.A. Yorke, An open set of maps for which every point is absolutely nonshadowable, Proc. Amer. Math. Soc. 128 (2000), 909–918.