A two-parameter analysis of the interaction between two period-doubling modes with azimuthal wavenumbers k and l (l>k≥1) is carried out for systems with circular symmetry. The problem is formulated in terms of an O(2)-equivariant map on C2. In the generic case all primary, secondary and tertiary solution branches and their stability properties are classified. The results depend on whether k + l is odd or even. When one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection-symmetric mixed modes, which in turn can undergo a tertiary Hopf bifurcation to a quasiperiodic reflection-symmetric pattern. We conjecture that this invariant circle can break down with increasing amplitude to produce reflection-symmetric chaos. Additional "phase" instabilities may occur in which the reflection symmetry is broken, and the resulting pattern drifts either clockwise or counterclockwise. The results explain a number of observations by Ciliberto and Gollub on parametrically excited surface waves in circular geometry, and imply several new predictions for such experiments. In an appendix the theory is compared with previous attempts to model the experiments.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics