Homoclinic structure of the dynamics of axisymmetric Bianchi type-IX universes and the oscillatory loitering phase

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Abstract
We study the chaotic dynamics of axisymmetric Bianchi type-IX universes from the point of view of its homoclinic structure. The presence of a cosmological constant engenders in the phase space of the model the topology of homoclinic cylinders emanating from the periodic orbits of the center manifold, connected to a critical point of saddle-center type. The nonintegrability of the dynamics causes the breaking and crossing of the cylinders, producing chaotic sets in the whole phase space of the model. The code associated with the two asymptotic configurations—escape into an inflationary phase or recollapse—defines fractal basin boundaries in initial conditions sets for the dynamics. These fractal boundaries are constituted by an uncountable set of homoclinic orbits and a countable set of periodic orbits with arbitrarily large periods. The fractal dimension is calculated using a box-counting procedure. We introduce an analytical technique to describe the nonlinear neighborhood of the saddle center, which provides suitable initial conditions for homoclinic orbits with their oscillatory approach to the unstable periodic orbits of the center manifold. In this treatment, new canonical variables are chosen such that the anisotropy may be interpreted as a scalar field with a complicated potential in an “average” Friedmann-Robertson-Walker universe. The neighborhood of the saddle center constitutes a loitering oscillatory phase for orbits visiting this neighborhood. We show that these oscillations provide a resonance mechanism for amplification of a selected wavelength spectrum of inhomogeneous perturbations in the models; and the form of the spectrum is evaluated.

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