14 research outputs found
Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity
In this paper we prove the breakdown of an heteroclinic connection in the
analytic versal unfoldings of the generic Hopf-Zero singularity in an open set
of the parameter space. This heteroclinic orbit appears at any order if one
performs the normal form around the origin, therefore it is a phenomenon
"beyond all orders". In this paper we provide a formula for the distance
between the corresponding stable and unstable one dimensional manifolds which
is given by an exponentially small function in the perturbation parameter. Our
result applies both for conservative and dissipative unfoldings
Differential Calculi on Associative Algebras and Integrable Systems
After an introduction to some aspects of bidifferential calculus on
associative algebras, we focus on the notion of a "symmetry" of a generalized
zero curvature equation and derive Backlund and (forward, backward and binary)
Darboux transformations from it. We also recall a matrix version of the binary
Darboux transformation and, inspired by the so-called Cauchy matrix approach,
present an infinite system of equations solved by it. Finally, we sketch recent
work on a deformation of the matrix binary Darboux transformation in
bidifferential calculus, leading to a treatment of integrable equations with
sources.Comment: 19 pages, to appear in "Algebraic Structures and Applications", S.
Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics,
202