No abstract

We study the dynamics of anisotropic Bianchi type-IX models with matter and cosmological constant. The models can be thought as describing the role of anisotropy in the early stages of inflation, where the cosmological constant Λ plays the role of the vacuum energy of the inflaton field. The concurrence of the cosmological constant and anisotropy are sufficient to produce a chaotic dynamics in the gravitational degrees of freedom, connected to the presence of a critical point of saddle-center type in the phase space of the system. In the neighborhood of the saddle-center, the phase space presents the structure of cylinders emanating from unstable periodic orbits. The non-integrability of the system implies that the extension of the cylinders away from this neighborhood has a complicated structure arising from their transversal crossings, resulting in a chaotic dynamics. The invariant character of chaos is guaranteed by the topology of cylinders. The model also presents a strong asymptotic de Sitter attractor but the way out from the initial singularity to the inflationary phase is completely chaotic. For a large set of initial conditions, even with very small anisotropy, the gravitational degrees of freedom oscillate a long time in the neighborhood of the saddle-center before recollapsing or escaping to the de Sitter phase. These oscillations may provide a resonance mechanism for amplification of specific wavelengths of inhomogeneous fluctuations in the models. A geometrical interpretation is given for Wald's inequality in terms of invariant tori and their destruction by increasing values of the cosmological constant.

This erratum corrects the interpretation we have made on the existence of homoclinic chaos in the model. The homoclinic structure constructed in the paper is based on the fact that A 0 is not a singularity of the Hamiltonian, and that the dynamics in the two regions of phase space A > 0 and A < 0 joins smoothly in A 0. However if the dynamics is restricted to A > 0, the essential recurrence is eliminated and the only boundary between initial conditions leading to collapse or escape is simply the manifold W S . The conclusions of the paper are therefore affected: the mathematical model is chaotic while its physical restriction is not. In this sense the formulation of Heinzle, Röhr and Uggla, Phys. Rev. D 71, 083506 (2005) does not detect chaos as their regularization prevents recurrence in the dynamics. They thus correctly establish the physical integrability of the system. PHYSICAL REVIEW D 73, 069901(E) (2006) 1550-7998= 2006=73(6)=069901(1)$23.00 069901-1

In this work we study the existence of mechanisms of the transition to global chaos in a closed Friedmann-RobertsonWalker universe with a massive conformally coupled scalar field. We propose a complexification of the radius of the universe so that the global dynamics can be understood. We show numerically the existence of heteroclinic connections of the unstable and stable manifolds to periodic orbits associated with the saddle-center equilibrium points. We find two bifurcations which are crucial in creating non-collapsing universes both in the real version and in the imaginary extension of the models. The techniques presented here can be employed in any cosmological model.

In this work we use a recently developed nonintegrability theorem of Morales and Ramis to prove that the Friedmann Robertson Walker cosmological model with a conformally coupled massive scalar field is nonintegrable.