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Chaos in the Hopf bifurcation with tetrahedral symmetry: Convection in a rotating fluid with low Prandtl number. (English) Zbl 0728.76049

Summary: We demonstrate that there are chaotic dynamics near the origin in the codimension-one Hopf bifurcation with tetrahedral symmetry. It has been observed by a number of authors, including the first author [Fluids and plasmas: geometry and dynamics, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 28, 435-448 (1984; Zbl 0535.76047)] and J. Guckenheimer and P. Holmes [Math. Proc. Camb. Philos. Soc. 103, No.1, 189-192 (1988; Zbl 0645.58022)], that tetrahedral symmetry acting on \({\mathbb{R}}^ 3\) can force homoclinic cycles in the normal form of the stationary bifurcation. We consider Hopf bifurcation, for which the tetrahedral symmetry acts on \({\mathbb{C}}^ 3\). In the case of stationary bifurcation the dynamics are simple in the neighbourhood of the heteroclinic cycle. In the Hopf case, however, numerical integrations show a chaotic attractor in the normal form near a saddle-focus heteroclinic cycle.
This bifurcation arises naturally in the problem of thermal convection in a rotating fluid layer with small Prandtl number. The conduction state loses stability to oscillations, in the form of standing or travelling waves. The problem we consider here is equivalent to three small- amplitude standing waves interacting on a hexagonal lattice.

MSC:

76E15 Absolute and convective instability and stability in hydrodynamic stability
76E30 Nonlinear effects in hydrodynamic stability
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
76U05 General theory of rotating fluids
37G99 Local and nonlocal bifurcation theory for dynamical systems
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