Hopf bifurcation with the symmetry of the square

Abstract

Four identical nonlinear oscillators, coupled with the symmetry of a square, can undergo o symmetric version of the standard Hopf bifurcation. Golubitsky and Stewart have studied the case of N oscillators coupled in a ring with nearest-neighbour coupling. Their results are incomplete for the square case ( N = 4) because they only considered periodic solutions which have 'maximal' symmetry. Here we study the dyncjmics of all possible square-symmetric Hopf bifurcations; these codimension-one bifurcations are parametrised by the three complex cubic coefficients in the normal form. We find that invariant tori (quasiperiodic solutions with two frequencies) and periodic solutions with 'minimal' symmetry bifurcate from the origin for open regions of the parameter space of cubic coefficients. The Coefficients can be chosen so that the invariant ton are the only asymptotically stable solutions near the origin. Thus a direct transition from a stable fixed point to flow on a stable invariant torus is expected in certain laboratory experiments with only one adiustable parameter (provided the square symmetry is accurate enough). Furthermore, it is conjectured that there are attracting chaotic solutions arbitrarily close to the bifurcation in a certain codimension-two case.

Oscillatory bifurcation with broken translation symmetry

“…Equations (18a,b) are identical to the equations describing a Hopf bifurcation with D4 symmetry considered by Swift [27] in the context of coupled oscillators. However, since our primary interest is in the large but finite L case (for particular scaling regimes), we are led to consider symmetry-breaking unfoldings of this limiting D4-symmetric case.…”

“…Consequently the behavior of each mode on the "fast" time scale is dominated by its oscillation frequency, Le., zl rv eiLl.w t, z2 rv e -iC.w t; cf. (27). Looking now at the nonlinear terms in (24a,b) it is clear that the final term in each fluctuates rapidly compared to the first two terms and hence effectively "averages out" to zero.…”

“…(24a,b) have the symmetry (34) in addition to the S1 normal form symmetry. To study these equations, the following coordinate transformation proves useful [27]: …”

A Note on Hopf Bifurcation With Dihedral Group Symmetry

“…Note that the case n = 4 and so m − 2 = 0 is special. The existence of branches of periodic solutions of (4.12) with submaximal symmetry that bifurcate from (0, 0) in a generic Hopf bifurcation with D 4 -symmetry is proved by Swift [8].…”

“…In this note we prove in Theorem 4.2 that if we assume (1.1) satisfying the conditions of the Equivariant Hopf Theorem and f is in Birkhoff normal form then, when n = 4 and n ≥ 3, the only branches of smallamplitude periodic solutions of period near 2π of (1.1) that bifurcate generically from the trivial equilibrium are the branches of solutions guaranteed by the Equivariant Hopf Theorem. The case when n = 4 differs markedly from those other n. Swift [8] studies the dynamics of all possible square-symmetric codimension one Hopf bifurcations. In particular, it is shown that periodic solutions with submaximal symmetry bifurcate from the origin for open regions of the parameter space of the cubic coefficients in the Birkhoff normal form.…”