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
Motivated by decoupling effects in coupled oscillators, by viscous shock profiles in systems of nonlinear hyperbolic balance laws, and by binary oscillation effects in discretizations of systems of hyperbolic balance laws, we consider vector fields with a one-dimensional line of equilibria, even in the absence of any parameters. Besides a trivial eigenvalue zero we assume that the linearization at these equilibria possesses a simple pair of nonzero eigenvalues which cross the imaginary axis transversely as we move along the equilibrium line.In normal form and under a suitable nondegeneracy condition, we distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Going beyond normal forms we present a rigorous analysis of both cases. In particular, :-and |-limit sets of nearby trajectories consist entirely of equilibria on the line.
We consider discretized systems of hyperbolic balance laws. Decoupling of the flow, associated with a central difference scheme, can lead to binary oscillations -even and odd numbered grid points, separately, provide time-evolutions of two distinct, different, separate profiles.Investigating the stability of this decoupling phenomenon, we encounter Hopf-like bifurcations in the absence of parameters. With some computer-algebra assistance, we describe the qualitative behavior near these bifurcation points. In particular we observe distinct even/odd profiles which oscillate periodically in time and, for arbitrarily fine discretization, exhibit preferred, nonzero phase relationships between adjacent discretization points.
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