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Bifurcations of Asymptotically Autonomous Differential Equations

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Abstract

This article deals with bifurcation phenomena of asymptotically autonomous differential equations. Under the assumption that the underlying autonomous system admits a bifurcation of pitchfork, saddle node, transcritical or Hopf type, nonautonomous bifurcation results are obtained for both the bifurcation of attraction and repulsion areas and transitions of attractors and repellers.

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References

  1. Abraham, R.H., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, New York (1988)

    MATH  Google Scholar 

  2. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    MATH  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Systems and Control: Foundations and Applications, vol. 2. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  4. Aulbach, B., Rasmussen, M., Siegmund, S.: Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete Contin. Dynam. Systems 15(2), 579–596 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berger, A., Siegmund, S.: Uniformly attracting solutions of nonautonomous differential equations. Nonlinear Anal. 68(12), 3789–3811 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cheban, D.N., Kloeden, P.E., Schmalfuß, B.: Pullback attractors in dissipative nonautonomous differential equations under discretization. J. Dynam. Differential Equations 13(1), 185–213 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fabbri, R., Johnson, R.A.: On a saddle-node bifurcation in a problem of quasi-periodic harmonic forcing, EQUADIFF 2003. In: Dumortier, F., Broer, H., Mawhin, J., Vanderbauwhede, A., Lunel, V. (eds.) Proceedings of the International Conference on Differential Equations, pp. 839–847. Hasselt, Belgium (2005)

  9. Fabbri, R., Johnson, R.A., Mantellini, F.: A nonautonomous saddle-node bifurcation pattern. Stochastics and Dynamics 4(3), 335–350 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Flandoli, F., Schmalfuß, B.: Random attractors for the 3-D stochastic Navier–Stokes equation with mulitiplicative white noise. Stochastics Stochastics Rep. 59(1–2), 21–45 (1996)

    MATH  MathSciNet  Google Scholar 

  11. Glendinning, P.: Non-smooth pitchfork bifurcations. Discrete Contin. Dynam. Systems B 4(2), 457–464 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Johnson, R.A.: An application of topological dynamics to bifurcation theory, topological dynamics and applications. In: Nerurkar, M.G., Dokken, D.P., Ellis, D.B. (eds.) A volume in honor of Robert Ellis, Contemporary Mathematics, vol. 215, pp. 323–334. American Mathematical Society, Providence, Rhode Island (1998)

    Google Scholar 

  13. Johnson, R.A., Kloeden, P.E., Pavani, R.: Two-step transition in nonautonomous bifurcations: an explanation. Stochastics and Dynamics 2(1), 67–92 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Johnson, R.A., Mantellini, F.: A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete Contin. Dynam. Systems 9(1), 209–224 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Kloeden, P.E.: Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure and Applied Analysis 3(2), 161–173 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kloeden, P.E., Siegmund, S.: Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems. Internat. J. Bifur. Chaos 15(3), 743–762 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Langa, J.A., Robinson, J.C., Suárez, A.: Stability, instability and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15(3), 887–903 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Langa, J.A., Robinson, J.C., Suárez, A.: Forwards and pullback behaviour of a non-autonomous Lotka–Volterra system. Nonlinearity 16(4), 1277–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Langa, J.A., Robinson, J.C., Suárez, A.: Bifurcations in non-autonomous scalar equations. J. Differential Equations 221(1), 1–35 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Markus, L.: Asymptotically autonomous differential systems. In: Lefschetz, S. (ed.) Contributions to the Theory of Nonlinear Oscillations III, Annals of Mathematical Studies, vol. 36, pp. 17–29. Princeton University Press (1956)

  21. Marsden, J.E., McCracken, M.: The Hopf Bifurcation and its Applications. Applied Mathematical Sciences, vol. 19. Springer, Berlin, Heidelberg, New York (1976)

    MATH  Google Scholar 

  22. Núñez, C., Obaya, R.: A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete Contin. Dynam. Systems A 9(3–4), 701–730 (2008)

    MATH  Google Scholar 

  23. Pötzsche, C.: Nonautonomous continuation and bifurcation of bounded solutions I: difference equations (to be submitted)

  24. Pötzsche, C.: Robustness of hyperbolic solutions under parametric perturbations. J. Differ. Equations Appl. (in press)

  25. Rasmussen, M.: Towards a bifurcation theory for nonautonomous difference equations. J. Differ. Equations Appl. 12(3–4), 297–312 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rasmussen, M.: Attractivity and Bifurcation for Nonautonomous Dynamical Systems. Springer Lecture Notes in Mathematics, vol. 1907. Springer, Berlin, Heidelberg, New York (2007)

    MATH  Google Scholar 

  27. Rasmussen, M.: Nonautonomous bifurcation patterns for one-dimensional differential equations. J. Differential Equations 234(1), 267–288 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)

    MATH  Google Scholar 

  29. Thieme, H.: Asymptotically autonomous differential equations in the plane. Rocky Mountain J. Math. 24(1), 351–380 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Augsburg, 86135, Augsburg, Germany

    Martin Rasmussen

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  1. Martin Rasmussen
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Correspondence to Martin Rasmussen.

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Research supported by the Bayerisches Eliteförderungsgesetz of the State of Bavaria and the Graduiertenkolleg Nichtlineare Probleme in Analysis, Geometrie und Physik (GK 283) financed by the DFG.

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Rasmussen, M. Bifurcations of Asymptotically Autonomous Differential Equations. Set-Valued Anal 16, 821–849 (2008). https://doi.org/10.1007/s11228-008-0089-5

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