Stable and non-symmetric pitchfork bifurcations
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1837–1852 https://doi.org/10.1007/s11425-019-1758-5
Stable and non-symmetric pitchfork bifurcations In Memory of Professor Shantao Liao
Enrique Pujals1 , Michael Shub2,∗ & Yun Yang3 1Department
of Mathematics, Graduate Center, City University of New York, New York, NY 10016, USA; of Mathematics, City College of New York, New York, NY 10031, USA; 3Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA 2Department
Email: [email protected], [email protected], [email protected] Received February 12, 2019; accepted August 6, 2020; published online August 20, 2020
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. Keywords MSC(2010)
pitchfork bifurcation, index, non-symmetric vector field, center manifold 34D10, 37C25, 37C20
Citation: Pujals E, Shub M, Yang Y. Stable and non-symmetric pitchfork bifurcations. Sci China Math, 2020, 63: 1837–1852, https://doi.org/10.1007/s11425-019-1758-5
1
Introduction
In this paper, we consider the bifurcation of isolated equilibria of locally defined vector fields in Rn . This well-studied subject has recently had some fresh observations by Rajapakse and Smale [15] concerning the pitchfork bifurcation and its relevance for biology. It is our intention to expand on their treatment by generalizing the hypotheses and uncovering significant new subtleties. First, let us recall the context: when locally defined vector fields and their bifurcations are used to model a phenomenon in the observable world, the fact that the phenomenon is observable at all speaks to its stability under small perturbation. The dogma of perturbation and bifurcation theory reasonably asserts that the aspects of the dynamics of the vector fields and their bifurcations used to explain the phenomenon should be stable as well. The only generic and stable simple non-hyperbolic bifurcation with one-dimensional parameters is the saddle-node bifurcation, in which zeros of adjacent indices are created or canceled. While the pitchfork bifurcation is not generally stable, it is stable under a certain additional hypothesis such as symmetry (namely equivariant branching) or the vanishing of a certain second derivative at the bifurcation point (see [12] and [9, Theorem 7.7]). The stability and the symmetry of the pitchfork bifurcation is usually expressed in terms of its normal form u˙ = uε − u3 . This family of vector fields is invariant under the involution u → −u. Rajapakse and Smale [13–15] were most interested in the case where one stable equilibrium gives rise to two new stable equilibria after the bifurcation and without * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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