Period-doubling mode interactions with circular symmetry
In an experiment of \textit{S. Ciliberto} and \textit{J. P. Gollub} [J. Fluid Mech. 158, 381-398 (1985)], a cylindrical container filled with fluid is oscillated vertically with frequency \(f_ 0\). When the amplitude A of the forcing exceeds a critical value, the surface of the fluid breaks up into a pattern of standing waves with temporal frequency \(f_ 0/2\). There are points in \((f_ 0,A)\)-space near which two such patterns, with different azimuthal wavenumbers k and l, occur. Ciliberto and Gollub studied in detail the mode interaction with \(k=4\) and \(l=7\). They observed, in addition to standing waves, a regime of periodic amplitude oscillation with a new frequency \(f^*\), and a chaotic regime. The present paper explains these experimental results as an instance of two simultaneous period doubling bifurcations in the presence of O(2) symmetry. The group O(2) is generated by rotations \(\phi: \theta \to \theta +\phi\) and a reflection \(\kappa: \theta \to -\theta.\) This group acts on \({\mathbb{C}}^ 2\) as follows \[ \phi \cdot (z,w)=(e^{ik\phi}z,e^{il\phi}w),\quad \kappa \cdot (z,w)=(\bar z,\bar w). \] The authors make a detailed study of maps f of \({\mathbb{C}}^ 2\) that are equivariant with respect to this representation of O(2) and have \(f(0)=0\), \(Df(0)=-I\). They identify period-doubling bifurcations, invariant circles and tori, and calculate their stability. The chaos observed experimentally is explained as due to a conjectured wrinkling of an invariant circle. The theory is in rather good agreement with the experimental results, except for several points pointed out by the authors: a mixed-mode state predicted by theory was not observed by the experimentalists; the observed chaotic regime seems larger than the predicted one; and the dimension of the chaotic attractor as computed by the experimentalists is larger than can be explained by the theory.
scientific article; zbMATH DE number 4157545
scientific article