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Stable and non-symmetric pitchfork bifurcations

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Abstract

In this paper, we present a criterion for pitchfork bifurcations of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale℉s result [15] significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.

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References

  1. Burns K, Gidea M. Differential Geometry and Topology with a View to Dynamical Systems. Boca Raton: Chapman and Hall/CRC, 2005

    Book  Google Scholar 

  2. Carr J. Applications of Center Manifold Theory. New York: Springer-Verlag, 1981

    Book  Google Scholar 

  3. Chow S N, Hale J K. Methods of Bifurcation Theory. New York: Springer, 1982

    Book  Google Scholar 

  4. Crandall J K, Rabinowitz P H. Bifurcation from simple eigenvalues. J Funct Anal, 1971, 8: 321–340

    Article  MathSciNet  Google Scholar 

  5. Crandall P H, Rabinowitz P H. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch Ration Mech Anal, 1973, 52: 161–180

    Article  MathSciNet  Google Scholar 

  6. Golubitsky M, Schaeffer D G. Singularities and Groups in Bifurcation Theory. Volume I. Applied Mathematical Sciences, vol. 51. New York: Springer, 1985

    Book  Google Scholar 

  7. Hirsch M W, Pugh P C, Shub M. Invariant Manifolds. New York: Springer, 2006

    MATH  Google Scholar 

  8. Kirchgassner K, Sorger P. Stability analysis of branching solutions of the Navier-Stokes equations. In: Applied Mechanics. International Union of Theoretical and Applied Mechanics. Berlin: Springer, 1969, 257–268

    Google Scholar 

  9. Kuznetsov Y. Elements of Applied Bifurcation Theory, 2nd ed. New York: Springer-Verlag, 1998

    MATH  Google Scholar 

  10. Liu P, Shi J, Wang Y. Imperfect transcritical and pitchfork bifurcations. J Funct Anal, 2007, 251: 573–600

    Article  MathSciNet  Google Scholar 

  11. Nirenberg L. Variational and topological methods in nonlinear problems. Bull Amer Math Soc (NS), 1981, 4: 267–302

    Article  MathSciNet  Google Scholar 

  12. Rabinowitz P H. Some global results for nonlinear eigenvalue problems. J Funct Anal, 1971, 7: 487–513

    Article  MathSciNet  Google Scholar 

  13. Rajapakse I, Smale S. Mathematics of the genome. Found Comput Math, 2017, 17: 1195–1217

    Article  MathSciNet  Google Scholar 

  14. Rajapakse I, Smale S. Emergence of function from coordinated cells in a tissue. Proc Natl Acad Sci USA, 2017, 114: 1462–1467

    Article  MathSciNet  Google Scholar 

  15. Rajapakse I, Smale S. The pitchfork bifurcation. Internat J Bifur Chaos, 2017, 27: 1750132

    Article  MathSciNet  Google Scholar 

  16. Ruelle D. Bifurcations in the presence of a symmetry group. Arch Ration Mech Anal, 1973, 51: 136–152

    Article  MathSciNet  Google Scholar 

  17. Sattinger D H. Stability of bifurcating solutions by Leray-Schauder degree. Arch Ration Mech Anal, 1971, 43: 154–166

    Article  MathSciNet  Google Scholar 

  18. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer, 2003

    MATH  Google Scholar 

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Acknowledgements

The second author was supported by the Smale Institute. This work was finished during the third author℉s stay in Graduate Center of City University of New York.

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Correspondence to Michael Shub.

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In Memory of Professor Shantao Liao

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Pujals, E., Shub, M. & Yang, Y. Stable and non-symmetric pitchfork bifurcations. Sci. China Math. 63, 1837–1852 (2020). https://doi.org/10.1007/s11425-019-1758-5

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  • DOI: https://doi.org/10.1007/s11425-019-1758-5

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