Chaos from Hopf bifurcation in a fluid flow experiment

Abstract

Results of an experimental study of a Hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a T2 torus in a fluid flow as a direct consequence of physical boundaries are presented. It is shown that the central features of the theory of Hopf bifurcation in O(2)-symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves, homoclinic bifurcation, Takens-Bogdanov point, and chaotic dynamics in a fluid flow experiment.

Bifurcation behavior of standing waves

“…A Taylor-Couette apparatus with counter-rotating cylinders and non-rotating end plates is used for this study, as described in [1][2][3]. The inner cylinder of the apparatus is machined from stainless steel having a radius of r i = (12.50 ± 0.01) mm, while the outer cylinder is made from optically polished glass with a radius of r o = (25.00 ± 0.01) mm.…”

“…While the former part has been observed in experiments for both types of standing waves, i.e., SW 0 and SW π [1,2], the latter point has only been investigated for SW 0 [3]. In this work we focus on higher bifurcations from the other type of standing wave in counter-rotating Taylor-Couette flow, i.e., SW π .…”

“…Two different types of standing waves (SW0 and SWπ) can appear instead of spiral vortices from a supercritical Hopf bifurcation in counter-rotating Taylor-Couette flow for sufficiently small aspect ratios [1,2]. The bifurcation sequence from basic flow to spiral vortices via SW0 can include modulated waves, homoclinic bifurcations, and hysteresis as a consequence of broken translational invariance [3]. Here we show that the same kind of sequence can also occur for the other type of standing wave, i.e., SWπ.…”

Simple environment for developing methods of controlling chaos in spatially distributed systems

“…The origin of this phenomenon in nonlinear dynamic systems is connected with their sensitivity to initial conditions. In fact, chaotic behavior is a common phenomenon discovered in physics (liquids (Procaccia and Meron, 1986;Langenberg et al, 2004), lasers (Yamada and Graham, 1980;Used and Martin, 2010), plasma (Held et al, 1984;Banerjee et al, 2010)), mechanics (Chui and Ma, 1982;Ott, 2002), electronics (Gunaratne et al, 1989;Yim et al, 2004), chemistry (Argoul et al, 1987;Córdo-ba et al, 2006), biology (action of human heart (Govindan et al, 1998) and brain (Andrzejak et al, 2001;Gautama et al, 2003)), and in many other branches of science and industry.…”

Efficiency analysis of control algorithms in spatially distributed systems with chaotic behavior

“…Deterministic chaos defined as a complex, disordered behavior generated by nonlinear systems (Ott, 2002) is a widely studied phenomenon which has been discovered and observed in practice in many different areas like liquids (Procaccia and Meron, 1986;Langenberg et al, 2004), lasers (Yamada and Graham, 1980;Used and Martin, 2010), plasma (Held et al, 1984;Banerjee et al, 2010), mechanics (Chui and Ma, 1982;Ott, 2002), electronics (Gunaratne et al, 1989;Yim et al, 2004), chemistry (Argoul et al, 1987;Cordoba et al, 2006), biology-the action of the human heart (Govindan et al, 1998) and the brain (Andrzejak et al, 2001;Gautama et al, 2003), and in many others.…”