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
ABSTRACT. Functional differential equations with forward and backward delays arise naturally, for instance, in the study of travelling waves in lattice equations and as semi-discretizations of partial differential equations (PDEs) on unbounded domains. Linear functional differential equations of mixed type are typically ill-posed, i.e., there exists, in general, no solution to a given initial condition. We prove that Fredholm properties of these equations imply the existence of exponential dichotomies. Exponential dichotomies can be used in discretized PDEs and in lattice differential equations to construct multi-pulses, to perform Evans-function type calculations, and to justify numerical computations using artificial boundary conditions.
The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold.In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in the small viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.