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Multidimensional Dynamical Systems

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Sharkovsky Ordering

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Abstract

It is well known that appearance of the Sh-ordering is strongly related to the one-dimensionality of the phase space. It is natural to look for special classes of multidimensional maps (and, possibly, objects more complex than cycles) for which the Sh-ordering, or a modification thereof, takes place.

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References

  1. Alsedà, L., Llibre, J., Misiurewicz, M.: Combinatonal dynamics and entropy in dimension one. In: Advanced Series in Nonlinear Dynamics, vol. 5, 2nd edn. World Scientific, River Edge, NJ (2000)

    Google Scholar 

  2. AlSharawi, Z., Angelos, J., Elaydi, S., Rakesh, L.: An extension of Sharkovsky’s theorem to periodic difference equations. J. Math. Anal. Appl. 316(1), 128–141 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andres, J., Pastor, K.: A version of Sharkovskii’s theorem for differential equations. Proc. Amer. Math. Soc. 133(2), 449–453 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andres, J., Pastor, K., Šnyrychová, P.: A multivalued version of Sharkovskii’s theorem holds with at most two exceptions. J. Fixed Point Theory Appl. 2(1), 153–170 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andres, J., Fürst, T., Pastor, K.: Full analogy of Sharkovsky’s theorem for lower semicontinuous maps. J. Math. Anal. Appl. 340(2), 1132–1144 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balibrea, F., Linero, A.: Periodic structure of \(\sigma \)-permutation maps on \(I^n\). Aequationes Math. 62(3), 265–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balibrea, F., Linero, A.: Periodic structure of \(\sigma \)-permutation maps. II. The case \(T^n\). Aequationes Math. 64(1-2), 34-52 (2002)

    Google Scholar 

  8. Balibrea, F., Linero, A.: On the periodic structure of delayed difference equations of the form \(x_n=f(x_{n-k})\) on \(I\) and \(S^1\). J. Diff. Eq. Appl. 9(3–4), 359–371 (2003)

    Article  MATH  Google Scholar 

  9. Block, L.: Stability of periodic orbits in the theorem of Sharkovskii. Proc. Amer. Math. Soc. 81, 333–336 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Gidea, M., Zgliczynski, P., : Covering relations for multidimensional dynamical systems. I, II. J. Diff. Eq. Appl. 202, 32–58 (2004)

    Google Scholar 

  11. Kloeden, P.E.: On Sharkovsky’s cycle coexisting ordering. Bull. Austral. Math. Soc. 20, 171–177 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kolyada, S.F.: On dynamics of triangular maps of the square. Ergod. Th. & Dyn. Syst. 12, 749–768 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, M.-Ch., Zgliczynski, P.: On stability of forcing relations for multidimensional perturbations of interval maps. Fund. Math. 206, 241–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Romanenko, EYu.: Randomness in deterministic continuous-time difference equations. J. Diff. Eq. Appl. 16(2–3), 243–268 (2010)

    Google Scholar 

  15. Romanenko, EYu., Sharkovsky, A.N., Vereikina, M.B.: Self-structuring and self-similarity in boundary value problems. Int. J. Bifurcation & Chaos 5(5), 1407–1418 (1995)

    Google Scholar 

  16. Sharkovsky, O. M.: Dynamical systems generated by boundary value problems. Ideal turbulence. Computer turbulence (in Ukrainian). In: Proceedings of Ukrainian Math. Congress, Vol. 12, pp. 125–149. Kiev, Ukraine (2003)

    Google Scholar 

  17. Sharkovsky, A.N.: Difference equations and boundary value problems. In: New Progress in Difference Equations. Proceedings of ICDEA 2001, pp. 3–22. Taylor & Francis, Boca Raton, FL (2004)

    Google Scholar 

  18. Sharkovsky, A.N., Romanenko, EYu.: Ideal turbulence: Attractors of deterministic systems may lie in the space of random fields. Int. J. Bifurcation & Chaos 2(1), 31–36 (1992)

    Google Scholar 

  19. Sharkovsky, A.N.: Ideal turbulence. Nonlinear Dynamics 44, 15–27 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sharkovsky, A.N.: Universal phenomena in some infinite-dimensional dynamical systems. Int. J. Bifurcation & Chaos 5(5), 1419–1425 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sharkovsky, A. N.; aǐstrenko, Yu. L.; Romanenko, E. Yu.: Difference equations and their applications. Translated from the 1986 Russian original by D. V. Malyshev, P. V. Malyshev and Y. M. Pestryakov. Mathematics and its Applications, 250. Kluwer Academic Publishers Group, Dordrecht (1993)

    Google Scholar 

  22. Sharkovsky, A.N., Sivak, A.G.: Universal order and universal rate of solution bifurcations of difference-differential equations (in Russian). In: Approximate and Qualitative Methods of the Theory of Differential and Functional-Differential Equations, pp. 98–106. Inst. Mat., Ukrain. Akad. Nauk, Kiev (1983)

    Google Scholar 

  23. Sharkovsky, A.N., Sivak, A.G.: Universal phenomena in solution bifurcations of some boundary value problems. Nonlinear Math. Physics 1(2), 147–157 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stefankova, M.: The Sharkovsky program of classification of triangular maps - A Survey. Topol. Proc. 48, 135–150 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Zgliczynski, P.: Fixed point index for iterations of maps, topological horseshoe and chaos. Topological Methods in Nonlinear Analysis 8(1), 169–177 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zgliczynski, P.: Sharkovskii’s theorem for multidimensional perturbations of one-dimensional maps. Ergod. Th. & Dyn. Syst. 19(6), 1655–1684 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zgliczynski, P.: Sharkovskii’s theorem for multidimensional perturbations of one-dimensional maps II. Topological Methods in Nonlinear Analysis 14, 169–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zgliczynski, P.: Multidimensional perturbations of one-dimensional maps and stability of Sharkovskii ordering. Int. J. Bifur. & Chaos 9(9), 1867–1876 (1999)

    Article  MATH  Google Scholar 

  29. Zgliczynski, P.: On periodic points for systems of weakly coupled 1-dim maps. Nonlinear Anal. TMA 46(7), 1039–1062 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zgliczynski, P.: Computer assisted proof of chaos in the Henon map and in the Rossler equations. Nonlinearity 10, 243–252 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, USA

    Alexander M. Blokh

  2. Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    Oleksandr M. Sharkovsky

Authors
  1. Alexander M. Blokh
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  2. Oleksandr M. Sharkovsky
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Correspondence to Alexander M. Blokh .

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Blokh, A.M., Sharkovsky, O.M. (2022). Multidimensional Dynamical Systems. In: Sharkovsky Ordering. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-99125-8_4

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