Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

Abstract

We consider cosmological models of Bianchi type. In particular, we are interested in the alpha-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity. We prove the existence of a codimension-one family of solutions that limit, for t to negative infinity, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.Comment: Minor corrections and clarifications throughout the text, according to referee's suggestion

Spike statistics

“…Another possibility is to focus on the state space picture in [9,10], where [9] give some examples of so-called concatenated chains, where axes permutations are not quoted out (for a discussion, see [10]). In particular, one can explore periodic spike chains, which are the recurring spike analogues of the so-called heteroclinic cycles in the BKL case, which recently have attracted attention [31,32].…”

Bifurcation without Parameters

“…Examples include decoupling in networks of coupled oscillators [4,52], oscillatory viscous in inviscid profiles in hyperbolic balance laws [26,39], binary oscillations in discretizations [30], population dynamics [24], Bianchi cosmological models [40,55], stationary profiles in fluid flows [1,2], memristor dynamics [64], and many more. Chapters 6, 7, 13, 14 are devoted to particular applications.…”

“…Rigorous results on this correspondence have been missing for 40 years. First answers are given in [10,55,56], albeit excluding the neighborhoods of the Taub points. Questions on the passage near the bifurcation points and the global return of trajectories are still open.…”

Untitled

“…It has been argued that the map which associates to each bounce the nearest point on the Kasner circle possesses chaotic features; this is at the origin of the mixmaster behavior terminology, sometimes used in this context. Major progress concerning this issue has been achieved in [199,28,246], where existence of orbits exhibiting The following result of Ringström [248] provides further insight into the geometry of Bianchi IX space-times: Note that the observation of curvature blow-up provides a proof, alternative to that of [111], of the nonexistence of Cauchy horizons in generic Bianchi IX models.…”