Hopf bifurcation on a sphere

Abstract

Using the general theory of Hopf bifurcation with symmetry we study here the example where the group of symmetries is O(3), the rotations and reflections of a sphere. We make some amendments to previously published lists of C-axial isotropy subgroups of O(3) × S 1 and list the isotropy subgroups with fourdimensional fixed-point subspaces. We then study the particular example where O(3)×S 1 acts on the space V 3 ⊕V 3 where V 3 is the space of spherical harmonics of degree three. We find that in this case there are six C-axial isotropy subgroups of O(3)×S 1 . The equivariant Hopf theorem guarantees the existence of periodic solutions with each of these symmetries in O(3) × S 1 equivariant differential equations. Three of the solutions are found to be standing waves and the other three are travelling waves. We compute conditions for each of these solution branches to be stable and by restricting the O(3) × S 1 equivariant differential equations to four-dimensional invariant subspaces we are able to find additional periodic and quasiperiodic solutions.

Standing and travelling waves in a spherical brain model: The Nunez model revisited

“…Thus the symmetries of the branches of periodic solutions bifurcating at a dynamic instability where the modes of degree $n_c$ become unstable correspond to the subgroups $\Sigma \subset O\left(3\right)\times S^1$ which are $\mathrm{double-struckC}$-axial under the action of $O\left(3\right)\times S^1$ on $V_{n_c}\oplus V_{n_c}$. Which subgroups are $\mathrm{double-struckC}$-axial isotropy subgroups depends on the value of $n_c$ and have been determined for all values of $n_c$   [44], [45]. The isotropy subgroups of $O\left(3\right)\times S^1$ are twisted subgroups $$H^{\theta }=\{(h,\theta \left(h\right))\in O\left(3\right)\times S^1:h\in H\}\text{,}$$ where $H$ is a subgroup of $O\left(3\right)$ and $\theta :H\to S^1$ is a group homomorphism.…”

“…In this case Hopf bifurcations are not possible. The focus of the remainder of this paper will be on the emergence of spatio-temporal patterns as expected from the general theory of Hopf bifurcations with symmetry [44] , [45] .…”

“… , or ). The -axial isotropy for values of between 1 and 6 is given in Table 1 which is reproduced from [45] . All notations for the subgroups of here and throughout are consistent with that in [44] , [45] .…”

Analysis of dynamics of neural fields and neural networks

“…Here l corresponds to the order of the spherical harmonics, which span the eigenspace of the bifurcation. We summarise the conditions for the stability of certain periodic orbits, or branches, in terms of the normal form coefficients, as given by Iooss and Rossi [1989] and Sigrist [2010b]. Using the homological equation for the Hopf bifurcation, we derive analytic expressions for the normal form coefficients of these bifurcations.…”

“…However, by the equivariant Hopf theorem we have the guarantee that at least the maximal branches exist. To determine the stability of these branches, conditions on the normal forms have been derived by Iooss and Rossi [1989] for the l = 2 and by Sigrist [2010b] for the l = 3. In this section, we summarise their main results.…”