Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.

[1]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[2]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[3]  David A. Rand,et al.  Dynamics and symmetry. Predictions for modulated waves in rotating fluids , 1982 .

[4]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[5]  H. Swinney,et al.  Instabilities and transition in flow between concentric rotating cylinders , 1981 .

[6]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[7]  G. Taylor.  Stability of a Viscous Liquid Contained between Two Rotating Cylinders , 1923 .

[8]  Ian Stewart,et al.  Symmetry and stability in Taylor Couette flow , 1986 .

[9]  Martin Golubitsky,et al.  Pattern formation and bistability in flow between counterrotating cylinders , 1988 .

[10]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[11]  D. Ruelle.  Bifurcations in the presence of a symmetry group , 1973 .

[12]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[13]  G. Iooss.  Bifurcation and transition to turbulence in hydrodynamics , 1984 .

[14]  Michael Renardy,et al.  Bifurcation from Rotating Waves , 1982 .

[15]  M. Golubitsky,et al.  Primary instabilities and bicriticality in flow between counter-rotating cylinders , 1988 .

[16]  W. Langford.  Periodic and Steady-State Mode Interactions Lead to Tori , 1979 .

[17]  Martin Golubitsky,et al.  Classification and Unfoldings of Degenerate Hopf Bifurcations , 1981 .

[18]  William F. Langford,et al.  Interactions of Hopf and Pitchfork Bifurcations , 1980 .

[19]  P. Holmes.  UNFOLDING A DEGENERATE NONLINEAR OSCILLATOR: A CODIMENSION TWO BIFURCATION * , 1980 .

[20]  John Guckenheimer,et al.  On a codimension two bifurcation , 1981 .

[21]  M. Golubitsky,et al.  Hopf Bifurcation in the presence of symmetry , 1984 .