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Hopf Bifurcation in the presence of symmetry

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Abstract

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R 2, O(n) on R n, and O(3) in any irreducible representation on spherical harmonics.

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The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston

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Golubitsky, M., Stewart, I. Hopf Bifurcation in the presence of symmetry. Arch. Rational Mech. Anal. 87, 107–165 (1985). https://doi.org/10.1007/BF00280698

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