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Nonautonomous Bifurcations in Nonlinear Impulsive Systems

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Abstract

In this paper, we study existence of the bounded solutions and asymptotic behavior of an impulsive Bernoulli equations. Nonautonomous pitchfork and transcritical bifurcation scenarios are investigated. An examples with numerical simulations are given to illustrate our results.

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References

  1. Akhmet, M.: Principle of discontinuous dynamical systems. Springer-Verlag, New York (2010)

    Book  Google Scholar 

  2. Akhmet, M.U., Arugaslan, D., Beklioglu, M.: Impulsive control of the population dynamics. In: Agarval, R.P., Perera, K. (eds.) Proceedings of the conference on differential and difference equations, differential & difference equations and applications, pp. 21–30. Hindawi Publishing Corporation, Cairo (2006)

  3. Akhmet, M.U., Kashkynbayev, A.: Non-autonomous bifurcation in impulsive systems. Electr. J. Qual. Theory Differ. Equ. 2013, 74 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Akhmet, M.U., Kashkynbayev, A.: Nonautonomous transcritical and pitchfork bifurcations in impulsive systems. Miskolc Math. Notes 14, 737–748 (2013)

    Article  MathSciNet  Google Scholar 

  5. Akhmet, M.U., Kashkynbayev, A.: Finite-time nonautonomous bifurcation in impulsive system. Proc. 10QTDE Electronic Journal of Qualitative Theory of Differential Equations, Electronic Journal of Qualitative Theory of Differential Equations (accepted)

  6. Akhmet, M.U., Yilmaz, E.: Impulsive Hopfield type neural network systems with piecewise constant argument. Nonlinear Anal. Real World Appl. 11, 2584–2593 (2010)

    Article  MathSciNet  Google Scholar 

  7. Arnold, L.: Random dynamical systems. Springer, Berlin (1998)

    Book  Google Scholar 

  8. Aulbach, B., Rasmussen, M., Siegmund, S.: Invariant manifolds as pullback attractors of nonautonomous differential equations. Discret. Contin. Dyn. Syst. Ser. A 15, 579–596 (2006)

    Article  MathSciNet  Google Scholar 

  9. Caraballo, T., Langa, J.A.: On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn. Contin. Discret. Impulsive Syst. Ser. A Math. Anal. 10, 491–513 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Cheban, D.N.: Global attractors of non-autonomous dissipative dynamical systems. World Scientific, Singapore (2004)

    Book  Google Scholar 

  11. Cheban, D.N., Kloeden, P.E., Schmalfuss, B.: The relationship between pullback, forwards and global attractors of nonautonomous dynamical system. Nonlinear Dyn. Syst. Theory 2, 9–28 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Chepyzhov, V.V., Vishik, M.I.: Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl. 73, 279–333 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Crauel, H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. 63, 413–427 (2001)

    Article  MathSciNet  Google Scholar 

  14. Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 397–441 (1997)

    Article  MathSciNet  Google Scholar 

  15. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Prob. Theory Relat. Fields 100, 365–393 (1994)

    Article  MathSciNet  Google Scholar 

  16. Kloeden, P.E.: Pitchfork and transcritical bifurcation in systems with homogeneous nonlinearities and an almost periodic time coefficients. Commun. Pure Appl. Anal. 3, 161–173 (2004)

    Article  MathSciNet  Google Scholar 

  17. Kloeden, P.E., Schmalfuss, B.: Asymptotic behavior of non autonomous difference inclusion. Syst. Control Lett. 33, 275–280 (1998)

    Article  Google Scholar 

  18. Kloeden, P.E., Stonier, D.: Cocycle attractors of nonautonomously perturbed differential equations. Dyn. Discret. Contin. Impulsive Syst. 4, 221–226 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Kloeden, P.E., Rasmussen, M.: Nonautonomous dynamical systems. AMS, Providence (2011)

    Book  Google Scholar 

  20. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  21. Langa, J.A., Robinson, J.C.: A finite number of point observations which determine a non-autonomous fluid flow. Nonlinearity 14, 673–682 (2001)

    Article  MathSciNet  Google Scholar 

  22. Langa, J.A., Robinson, J.C., Suarez, A.: Stability, Instability and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15, 1–17 (2002)

    Article  MathSciNet  Google Scholar 

  23. Langa, J.A., Robinson, J.C., Suarez, A.: Forwards and pullback behavior of non-autonomous Lotka-Volterra system. Nonlinearity 16, 1277–1293 (2003)

    Article  MathSciNet  Google Scholar 

  24. Langa, J.A., Robinson, J.C., Suarez, A.: Bifurcation in non-autonomous scalar equations. J. Differ. Equ. 221, 1–35 (2006)

    Article  MathSciNet  Google Scholar 

  25. Langa, J.A., Suarez, A.: Pullback performance for non-autonomous partial differential equations. Electr. J. Differ. Equ. 2002(72), 1–20 (2002)

    MATH  Google Scholar 

  26. Nguyen, T.Y., Doan, T.S., Jager, T., Siegmund, S.: Nonautonomous saddle-node bifurcations in the quasiperiodically forced logistic map. Intern. J. Bifur. Chaos Appl. Sci. Eng. 5, 1427–1438 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Pötzsche, C.: Nonautonomous bifurcation of bounded solutions I: a Lyapunov-Schmidt approach. Discret. Contin. Dyn. Syst. Ser. B 14(2), 739–776 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Pötzsche, C.: Nonautonomous bifurcation of bounded solutions II: a shovel-bifurcation pattern. Discret. Contin. Dyn. Syst. Ser. A 31(3), 941–973 (2011)

    Article  MathSciNet  Google Scholar 

  29. Pötzsche, C.: Nonautonomous bifurcation of bounded solutions III: crossing-curve situations. Stoch. Dyn. 12(2), 1150017 (2012)

    Article  MathSciNet  Google Scholar 

  30. Rasmussen, M.: Attractivity and Bifurcation for nonautonomous dynamical systems. Springer-Verlag, Berlin (2007)

    MATH  Google Scholar 

  31. Samoilenko, A.M., Perestyuk, N.A.: Impulsive differential equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  32. Schmalfuss, B.: Attractors for non-autonomous dynamical system. In: Fiedler, B., Grger, K., Sprekels, J. (eds.) Proc. Equadiff 99 (Berlin), pp. 684–689. World Scientific, Singapore (2000)

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Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Marat Akhmet & Ardak Kashkynbayev

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  1. Marat Akhmet
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  2. Ardak Kashkynbayev
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Correspondence to Marat Akhmet.

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Akhmet, M., Kashkynbayev, A. Nonautonomous Bifurcations in Nonlinear Impulsive Systems. Differ Equ Dyn Syst 28, 177–190 (2020). https://doi.org/10.1007/s12591-016-0309-7

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