TY - JOUR

T1 - Period-doubling mode interactions with circular symmetry

AU - Crawford, John David

AU - Knobloch, Edgar

AU - Riecke, Hermann

N1 - Funding Information:
by NSF under grant DMS-8814702. H.R. was also supported by the Deutsche Forshungsgemeinschaft (DFG).
Funding Information:
We are grateful to Professors S. Ciliberto, J. Gollub, M. Golubitsky, and A. Vanderbauwhede for helpful discussions, and to Professor J. Gollub for providing the experimental diagram in fig. 1. This research was supported by DARPA under the Applied and Computational Mathematics Program, and

PY - 1990/9/1

Y1 - 1990/9/1

N2 - 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.

AB - 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.

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U2 - 10.1016/0167-2789(90)90153-G

DO - 10.1016/0167-2789(90)90153-G

M3 - Article

AN - SCOPUS:0001494072

SN - 0167-2789

VL - 44

SP - 340

EP - 396

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 3

ER -