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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burns K, Gidea M. Differential Geometry and Topology with a View to Dynamical Systems. Boca Raton: Chapman and Hall/CRC, 2005
Carr J. Applications of Center Manifold Theory. New York: Springer-Verlag, 1981
Chow S N, Hale J K. Methods of Bifurcation Theory. New York: Springer, 1982
Crandall J K, Rabinowitz P H. Bifurcation from simple eigenvalues. J Funct Anal, 1971, 8: 321–340
Crandall P H, Rabinowitz P H. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch Ration Mech Anal, 1973, 52: 161–180
Golubitsky M, Schaeffer D G. Singularities and Groups in Bifurcation Theory. Volume I. Applied Mathematical Sciences, vol. 51. New York: Springer, 1985
Hirsch M W, Pugh P C, Shub M. Invariant Manifolds. New York: Springer, 2006
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
Kuznetsov Y. Elements of Applied Bifurcation Theory, 2nd ed. New York: Springer-Verlag, 1998
Liu P, Shi J, Wang Y. Imperfect transcritical and pitchfork bifurcations. J Funct Anal, 2007, 251: 573–600
Nirenberg L. Variational and topological methods in nonlinear problems. Bull Amer Math Soc (NS), 1981, 4: 267–302
Rabinowitz P H. Some global results for nonlinear eigenvalue problems. J Funct Anal, 1971, 7: 487–513
Rajapakse I, Smale S. Mathematics of the genome. Found Comput Math, 2017, 17: 1195–1217
Rajapakse I, Smale S. Emergence of function from coordinated cells in a tissue. Proc Natl Acad Sci USA, 2017, 114: 1462–1467
Rajapakse I, Smale S. The pitchfork bifurcation. Internat J Bifur Chaos, 2017, 27: 1750132
Ruelle D. Bifurcations in the presence of a symmetry group. Arch Ration Mech Anal, 1973, 51: 136–152
Sattinger D H. Stability of bifurcating solutions by Leray-Schauder degree. Arch Ration Mech Anal, 1971, 43: 154–166
Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer, 2003
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
In Memory of Professor Shantao Liao
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-019-1758-5