Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2127–2144 | Cite as

Application of the Randomized Sharkovsky-Type Theorems to Random Impulsive Differential Equations and Inclusions

  • Jan AndresEmail author


Our randomized Sharkovsky-type theorems are applied to random impulsive differential equations and inclusions in order to establish the coexistence of random periodic solutions with various periods, which are forced according to the Sharkovsky ordering of positive integers. The impulses can be single-valued or multivalued and deterministic or random. The obtained theorems can be rather curiously stronger than their deterministic analogies. The relationship to deterministic chaos is also indicated.


Sharkovsky-type theorems Random differential equations and inclusions Deterministic and random impulses Coexistence of random periodic solutions Random chaos 

Mathematics Subject Classification

34B37 37E15 47H10 47H40 


  1. 1.
    Agarwal, R., Hristova, S., O’Regan, D.: Exponential stability for differential equations with random impulses at random times. Adv. Differ. Equ. 2013(372), 1–12 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andres, J.: Randomization of Sharkovskii-type theorems. Proc. Am. Math. Soc. 136(4), 1385–1395 (2008). Erratum, Proc. Am. Math. Soc. 136, 3733–3734 (2008)Google Scholar
  3. 3.
    Andres, J.: Randomization of Sharkovskii-type results on the circle. Stoch. Dyn. 17(3), 1750017 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andres, J.: On the notion of random chaos. Proc. Am. Math. Soc. 145, 3423–3435 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Andres, J., Barbarski, P.: Random Sharkovsky-type results and random subharmonic solutions of differential inclusions. Proc. Am. Math. Soc. 144, 1971–1983 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Andres, J., Fišer, J.: Sharkovsky-type theorems on ${S}^1$ applicable to differential equations. Int. J. Bifurc. Chaos 27(1750042), 1–21 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Andres, J., Fürst, T., Pastor, K.: Sharkovskii’s theorem, differential inclusions, and beyond. Topol. Methods Nonlinear Anal. 33, 149–168 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Andres, J., Górniewicz, L.: Topological Fixed Point Principles for Boundary Value Problems. Kluwer, Dordrecht (2003)CrossRefGoogle Scholar
  9. 9.
    Andres, J., Górniewicz, L.: Random topological degree and random differential inclusions. Topol. Methods Nonlinear Anal. 40, 337–358 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Appell, J., De Pascale, E., Thái, N.H., Zabreĭko, P.P.: Multi-valued superpositions, Dissertationes Mathematicæ 345. Instytut Matematyczny, Polska Akademia Nauk, Warsaw (1995)Google Scholar
  11. 11.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)CrossRefGoogle Scholar
  12. 12.
    Bainov, D., Simeonov, P.: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66. Longman Scientific & Technical, Harlow (1993)zbMATHGoogle Scholar
  13. 13.
    Barbarski, P.: The Sharkovskiĭ theorem for spaces of measurable functions. J. Math. Anal. Appl. 373, 414–421 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Beg, I., Abbas, M.: Periodic points of random multivalued operators. Nonlinear Stud. 19, 87–92 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Beg, I., Abbas, M., Azam, A.: Periodic fixed points of random operators. Ann. Math. Inform. 37, 39–49 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Boyarsky, A.: Randomness implies order. J. Math. Anal. Appl. 76, 483–497 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Menlo Park (1989)zbMATHGoogle Scholar
  18. 18.
    Halmos, P.R.: Measure Theory. Springer, Berlin (1974)zbMATHGoogle Scholar
  19. 19.
    Han, X., Kloeden, P.E.: Random Ordinary Differential Equations and Their Numerical Solution. Springer, Berlin (2017)CrossRefGoogle Scholar
  20. 20.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, Vol. I: Theory, Mathematics and its Applications, vol. 419. Kluwer, Dordrecht (1997)CrossRefGoogle Scholar
  21. 21.
    Johnson, R., Nerurkar, M.: Controllability, Stabilization, and the Regulator Problem for Random Differential Systems. Memoirs of the AMS 646. Amer. Math. Soc, Providence (1998)zbMATHGoogle Scholar
  22. 22.
    Klünger, M.: Periodicity and Sharkovsky’s theorem for random dynamical systems. Stoch. Dyn. 1, 299–338 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, T.-Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82, 985–992 (1975)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mil’man, V.D., Myshkis, A.D.: Random impulses in linear dynamical systems. In: Approximate Methods for Solving Differential Equations. Publishing House of the Academy of Sciences of Ukrainian SSR, Kiev, pp. 64–81 (1963) (in Russian) Google Scholar
  25. 25.
    Perestyuk, N.A., Plotnikov, V.A., Samoilenko, A.M., Skripnik, N.V.: Differential Equations with Impulse Effects. Multivalued Right-Hand Sides with Discontinuities. De Gruyter Studies in Mathematics, vol. 40. De Gruyter, Berlin (2011)CrossRefGoogle Scholar
  26. 26.
    Ruan, J., Lin, W.: Chaos in a class of impulsive differential equation. Commun. Nonlinear Sci. Numer. Simul. 4, 165–169 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Samoilenko, A.M., Stanzhytskyi, O.: Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations. World Scientific Series on Nonlinear Science Series A, vol. 78. World Scientific Publ, Singapore (2011)CrossRefGoogle Scholar
  28. 28.
    Sanz-Serna, J.M., Stuart, A.M.: Ergodicity of dissipative differential equations subject to random impulses. J. Differ. Equ. 155, 262–284 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sharkovsky, A.N.: Coexistence of cycles of a continuous map of the line into itself. Int. J. Bifurc. Chaos 5, 1263–1273 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Song, X., Guo, H., Shi, X.: Theory and Application of Impulsive Differential Equations. Science Press, Beijing (2011)Google Scholar
  31. 31.
    Srivastava, S.M.: A Course on Borel Sets. Springer, Berlin (1998)CrossRefGoogle Scholar
  32. 32.
    Vellekoop, M., Berglund, R.: On intervals, transitivity = chaos. Am. Math. Mon. 101, 353–355 (1994)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wu, S.J., Zhou, B.: Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-Lipschitz conditions. Acta Math. Sin. 27, 519–536 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yang, T., Yang, L.-B., Yang, C.-M.: Impulsive control of Lorenz system. Phys. D Nonlinear Phenom. 110, 18–24 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Analysis and Applications of Mathematics, Faculty of SciencePalacký UniversityOlomoucCzech Republic

Personalised recommendations