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Heteroclinic Cycles and Modulated Travelling Waves in a System with D4 Symmetry

S. Campbell
P. Holmes
Journal / Anthology

Physica D
Year: 1992
Volume: 59
Page range: 52-78

We study a system of two complex ordinary differential equations with D4 symmetry describing the effects of the symmetry breaking O(2) to D4 on an interaction between Fourier modes with wavenumbers in the ratio 1 : 2. This change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag. Numerical simulations, planar analysis and Melnikov's method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the O(2) system. It is further shown that one of these cycles destabilizes to regular modulated traveling waves which (quasi-)periodically reverse their direction of propagation. A bifurcation diagram for this behavior and the subsequent destabilization of modulated traveling waves is established. The method of averaging is used to show that some of the quasiperiodic solutions of the O(2) system persist under the symmetry breaking, while others break into sets of periodic orbits.

*Mathematics > Calculus and Analysis > Differential Equations
*Mathematics > Calculus and Analysis > Dynamical Systems