A Phenomenological Approach to Normal Form Modeling: a Case Study in Laser Induced Nematodynamics

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films presented by G. Cipparrone, G. Russo, C. Versace, G. Strangi and V. Carbone allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, relying on the original model for chaos by cascade of gluing bifurcations proposed by A. Arneodo, P. Coullet and C. Tresser about twenty years before, does not have the central symmetry which one would expect for minimal dimensional model for chaos in nematics in view of the time series near the gluing bifurcation. What we show here is that the simplest truncated normal forms for gluing with the appropriate symmetry and minimal dimension do exhibit time signals that are embarrassingly similar to the ones that could be found using the above-mentioned experimental settings. It so happens that the gluing bifurcation scenario itself is only visible in limited parameter ranges, and that a substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion which easily produces chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach to attracting homoclinic pairs contains a rich behavior, minute noise can transform the complex damping into sustained chaos. As Leonid Shilnikov has taught us, combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations in dissipative systems. We see here on an example how this helps construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.