Bianchi IX dynamics in bouncing cosmologies: homoclinic chaos and the BKL conjecture

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We examine the dynamics of a Bianchi IX model with three scale factors on a 4-dim Lorentzian brane embedded in a 5-dim conformally flat empty bulk with a timelike extra dimension. The matter content is a pressureless perfect fluid restricted to the brane, with the embedding consistently satisfying the Gauss–Codazzi equations. The 4-dim Einstein equations on the brane reduce to a 6-dim Hamiltonian dynamical system with additional terms (due to the bulk–brane interaction) that avoid the singularity and implement nonsingular bounces in the model. We examine the complex Bianchi IX dynamics in its approach to the neighborhood of the bounce which replaces the cosmological singularity of general relativity. The phase space of the model presents (i) two critical points (a saddle-center–center and a center–center–center) in a finite region of phase space, (ii) two asymptotic de Sitter critical points at infinity, one acting as an attractor to late-time acceleration and (iii) a 2-dim invariant plane, which together organize the dynamics of the phase space. The saddle-center–center engenders in the phase space the topology of stable and unstable 4-dim cylinders R × S3, where R is a saddle direction and S3 is the center manifold of unstable periodic orbits, the latter being the nonlinear extension of the center–center sector. By a proper canonical transformation the degrees of freedom of the dynamics are separated into one degree connected with the expansion/contraction of the scales of the model, and two rotational degrees of freedom associated with the center manifold S3. The typical dynamical flow is thus an oscillatory mode about the orbits of the invariant plane. The stable and unstable cylinders are spanned by oscillatory orbits about the separatrix towards the bounce, leading to the homoclinic transversal intersection of the cylinders, as shown numerically in two distinct simulations. The homoclinic intersection manifold has the topology of R × S2 consisting of homoclinic orbits biasymptotic to the center manifold S3. This behavior defines a chaotic saddle associated with S3, indicating that the intersection points of the cylinders have the nature of a Cantor set with compact support S2. This is an invariant signature of chaos in the model. We discuss the connection between these properties of the dynamics, namely the oscillatory approach to the bounce together with its chaotic behavior, and analogous features present in the BKL conjecture in general relativity.