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The abelian Hopf $H$ mod $K$ theorem. (English) Zbl 1198.37088
The symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries are investigated. Main result is the abelian Hopf $H\phantom{\rule{0.166667em}{0ex}}\text{mod}\phantom{\rule{0.166667em}{0ex}}K$ theorem which gives necessary and sufficient conditions for when these $H\phantom{\rule{0.166667em}{0ex}}\text{mod}\phantom{\rule{0.166667em}{0ex}}K$ periodic solutions can occur by Hopf bifurcation when ${\Gamma }$ is a finite Abelian group. Examples of these results in the case when the symmetry group . ${\Gamma }={ℤ}_{l}×{ℤ}_{k}$ acts on ${ℝ}^{l}×{ℝ}^{k}$ by permutation of coordinates are given. In this case the $H\phantom{\rule{0.166667em}{0ex}}\text{mod}\phantom{\rule{0.166667em}{0ex}}K$ periodic solutions that are obtainable by a generic Hopf bifurcation are classified and it is shown that there exist families of $H\phantom{\rule{0.166667em}{0ex}}\text{mod}\phantom{\rule{0.166667em}{0ex}}K$ periodic solutions that cannot be obtained by Hopf bifurcation.
MSC:
 37G40 Symmetries, equivariant bifurcation theory 34C23 Bifurcation (ODE) 34C25 Periodic solutions of ODE