Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Volume 10, Issue 9, pp 4940--4953 http://dx.doi.org/10.22436/jnsa.010.09.34

Publication Date: September 22, 2017 Submission Date: May 13, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2178 Downloads
• 3727 Views

Authors

Risong Li - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Tianxiu Lu - Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People's Republic of China. - Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, People’s Republic of China. - Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, People’s Republic of China. Yu Zhao - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Hongqing Wang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Haihua Liang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China.

Abstract

In this paper, for a continuous semi-flow \(\theta\) on a compact metric space \(E\) with the asymptotic average-shadowing property (AASP), we show that if the almost periodic points of \(\theta\) are dense in \(E\) then \(\theta\) is multi-sensitive and syndetically sensitive. Also, we show that if \(\theta\) is a Lyapunov stable semi-flow with the AASP, then the space \(E\) is trivial. Consequently, a Lyapunov stable semi-flow with the AASP is minimal. Furthermore, we prove that for a syndetically transitive continuous semi-flow on a compact metric space, sensitivity is equivalent to syndetical sensitivity. As an application, we show that for a continuous semi-flow \(\theta\) on a compact metric space \(E\) with the AASP, if the almost periodic points of \(\varphi\) are dense in \(E\) then \(\theta\) is syndetically sensitive. {Moreover, we prove that for any continuous semi-flow \(\theta\) on a compact metric space, it has the AASP if and only if so does its inverse limit \((\widetilde{E}, \widetilde{\theta})\), and if only if so does its lifting continuous semi-flow \((\widehat{E}, \widehat{\theta})\). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.

Share and Cite

Keywords

• The asymptotic average-shadowing property
• strong ergodicity
• minimal point
• multi-sensitivity
• syndetical sensitivity
• Lyapunov stable

MSC

•  54H20
•  37B40
•  37D45

References

• [1] J. F. Alves, M. Carvalho, C. H. Vásquez , A variational principle for impulsive semiflows, J. Differential Equations, 259 (2015), 4229–4252.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] N. Aoki, K. Hiraide, Topological theory of dynamical systems, Recent advances, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1994)
  • Google Scholar


• [3] A. Arbieto, R. Ribeiro , Flows with the (asymptotic) average shadowing property on three-dimensional closed manifolds, Dyn. Syst., 26 (2011), 425–432.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] W. Basener, M. C. Sullivan, Periodic prime knots and topologically transitive flows on 3-manifolds, Topology Appl., 153 (2006), 1236–1240.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] A. Bielecki, J. Ombach, Dynamical properties of a perceptron learning process: structural stability under numerics and shadowing, J. Nonlinear Sci., 21 (2011), 579–593.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Blank, Small perturbations of chaotic dynamical systems, (Russian); translated from Uspekhi Mat. Nauk, 44 (1989), 3–28, Russian Math. Surveys, 44 (1989), 1–33.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York (1975)
  • View Article
  • MathSciNet
  • Google Scholar


• [8] R. Bowen, P. Walters , Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180–193.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] J. Buzzi , Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737–2754.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] C.-K. Chu, K.-S. Koo , Recurrence and the shadowing property, Topology Appl., 71 (1996), 217–225.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] C. Conley , Isolated invariant sets and the Morse index , CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I. (1978)
  • MathSciNet
  • Google Scholar


• [12] M. Denker, C. Grillenberger, K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Springer- Verlag, Berlin-New York (1976)
  • MathSciNet


• [13] T. Gedeon, M. Kuchta, Shadowing property of continuous maps, Proc. Amer. Math. Soc., 115 (1992), 271–279.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] R.-B. Gu, The asymptotic average-shadowing property and transitivity for flows, Chaos Solitons Fractals, 41 (2009), 2234–2240.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] R.-B. Gu, W.-J. Guo , The average-shadowing property and topological ergodicity for flows, Chaos Solitons Fractals, 25 (2005), 387–392.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] R.-B. Gu, Y.-Q. Sheng, Z.-J. Xia , The average-shadowing property and transitivity for continuous flows, Chaos Solitons Fractals, 23 (2005), 989–995.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] L.-F. He, Y.-H. Gao, F.-H. Yang, Some dynamical properties of continuous semi-flows having topological transitivity, Chaos Solitons Fractals, 14 (2002), 1159–1167.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] L.-F. He, G.-Z. Shan, Lifting and projecting continuous flows with the pseudo-orbit tracing property or the expansibility property, (Chinese); Acta Math. Appl. Sinica, 18 (1995), 1–7.
  • MathSciNet
  • Google Scholar
  • MATH


• [19] L.-F. He, Z.-H. Wang , Distal flows with pseudo-orbit tracing property , Chinese Sci. Bull., 39 (1994), 1936–1938.


• [20] L.-F. He, Z.-H. Wang, H. Li, Continuous semiflows with the shadowing property , (Chinese); Acta Math. Appl. Sinica, 19 (1996), 297–303.
  • MathSciNet
  • Google Scholar


• [21] L.-F. He, Z. Zhang, Chaos in the semi-flows and its inverse limit systems, Acta Math. Sci., 17 (1997), 46–51.
  • Google Scholar


• [22] B. Honary, A. Zamani Bahabadi , Orbital shadowing property, Bull. Korean Math. Soc., 45 (2008), 645–650.
  • Google Scholar


• [23] L.-L. Huang, D. Baleanu, G.-C. Wu, S.-D. Zeng, A new application of the fractional logistic map, Rom. J. Phys., 61 (2016), 1172–1179.
  • Google Scholar


• [24] L.-L. Huang, G.-C. Wu, M. M. Rashidi, W.-H. Luo, Chaos analysis of the nonlinear Duffing oscillators based on the new Adomian polynomials, J. Nonlinear Sci. Appl., 9 (2016), 1877–1881.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] K. Kato, Pseudo-orbits and stabilities of flows, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 5 (1984), 45–62.
  • MathSciNet
  • Google Scholar
  • MATH


• [26] M. Komuro, One-parameter flows with the pseudo-orbit tracing property, Monatsh. Math., 98 (1984), 219–253.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] P. Kościelniak, M. Mazur, Chaos and the shadowing property , Topology Appl., 154 (2007), 2553–2557.
  • View Article
  • Google Scholar


• [28] M. Kulczycki, P. Oprocha, Exploring the asymptotic average shadowing property, J. Difference Equ. Appl., 16 (2010), 1131–1140.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] M. Kulczycki, P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fund. Math., 212 (2011), 35–52.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] S.-H. Lee, H.-J. Koh, S.-H. Ku , Investigation of the stability via shadowing property , J. Inequal. Appl., 2009 (2009), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] C.-P. Li, A new method of determining chaos-parameter-region for the tent map, Chaos Solitons Fractals, 21 (2004), 863–867.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] R.-S. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2815–2823.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] R.-S. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45 (2012), 753–758.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] R.-S. Li , A note on decay of correlation implies chaos in the sense of Devaney, Appl. Math. Model., 39 (2015), 6705–6710.
  • View Article
  • MathSciNet
  • Google Scholar


• [35] R.-S. Li, Y.-M. Shi , Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, Abstr. Appl. Anal., 2014 (2014), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [36] M. Magdziarz, A. Weron, Ergodic properties of anomalous diffusion processes , Ann. Physics, 326 (2011), 2431–2443.
  • View Article
  • MathSciNet
  • Google Scholar


• [37] J.-H. Mai , Pointwise recurrent dynamical systems with pseudo-orbit tracing property, Northeast. Math. J., 12 (1996), 73–78.
  • MathSciNet
  • Google Scholar
  • MATH


• [38] J. Marklof , Entry and return times for semi-flows, Nonlinearity, 30 (2017), 810–824.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115–2126.
  • View Article
  • MathSciNet
  • Google Scholar


• [40] Y.-X. Niu, S.-B. Su , On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property , Chaos Solitons Fractals, 44 (2011), 429–432.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] S. Y. Pilyugin, Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations, 19 (2007), 747–775.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] S. Y. Pilyugin, K. Sakai , \(C^0\) transversality and shadowing properties, translated from Tr. Mat. Inst. Steklova, Din. Sist. i Optim., 256 (2007), 305–319, Proc. Steklov Inst. Math., 256 (2007), 290–305.
  • View Article
  • MathSciNet
  • Google Scholar


• [43] K. Sakai , Diffeomorphisms with the average-shadowing property on two-dimensional closed manifolds, Rocky Mountain J. Math., 3 (2000), 1129–1137.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] K. Sakai, Shadowing properties of L-hyperbolic homeomorphisms, Topology Appl., 112 (2001), 229–243.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [45] K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl., 131 (2003), 15–31.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [46] S. Smale, Differentiable dynamical systems , Bull. Amer. Math. Soc., 73 (1967), 747–817.
  • View Article
  • MathSciNet
  • Google Scholar


• [47] R. F. Thomas, Stability properties of one-parameter flows , Proc. London Math. Soc., 45 (1982), 479–505.
  • View Article
  • MathSciNet
  • Google Scholar


• [48] D. Thomine , Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory Dynam. Systems, 35 (2015), 1610–1657.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [49] D. Thomine, Local time and first return time for periodic semi-flows, Israel J. Math., 215 (2016), 53–98.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [50] X.-J. Tong, Design of an image encryption scheme based on a multiple chaotic map, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1725–1733.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [51] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems, Proc. Conf., North Dakota State Univ., Fargo, N.D., (1977), Lecture Notes in Math., Springer, Berlin, 668 (1978), 231–244.
  • MathSciNet
  • Google Scholar
  • MATH


• [52] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin (1982)
  • MathSciNet


• [53] G.-C. Wu, D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Process., 102 (2014), 96–99.
  • View Article
  • Google Scholar


• [54] G.-C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283–287.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [55] G.-C. Wu, D. Baleanu, H.-P. Xie, F.-L. Chen, Chaos synchronization of fractional chaotic maps based on the stability condition, Phys. A, 460 (2016), 374–383.
  • View Article
  • MathSciNet
  • Google Scholar


• [56] S.-L. Wu, G.-C. Wu, Nonoverlapping Schwarz waveform relaxation algorithm for a class of time-fractional heat equations, Fund. Inform., 151 (2017), 231–240.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [57] X.-H. Yan, L.-F. He , Two remarks on sensitive dependence of semi-dynamical systems, Southeast Asian Bull. Math., 32 (2008), 393–398.
  • MathSciNet
  • Google Scholar
  • MATH


• [58] R.-S. Yang , The pseudo-orbit tracing property and chaos, (Chinese); Acta Math. Sinica (Chin. Ser.), 39 (1996), 382–386.
  • Google Scholar


• [59] R. S. Yang, S. L. Shen, Pseudo-orbit-tracing and completely positive entropy, (Chinese); Acta Math. Sinica (Chin. Ser.), 42 (1999), 99–104.
  • MathSciNet
  • Google Scholar
  • MATH