Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

The next 700 Krivine Machines

International audienceThe Krivine machine is a simple and natural implementation of the normal weak-head reduction strategy for pure lambda-terms. While its original description has remained unpublished, this machine has served as a basis for many variants, extensions and theoretical studies. In this paper, we present the Krivine machine and some well-known variants in a common framework. Our framework consists of a hierarchy of intermediate languages that are subsets of the lambda-calculus. The whole implementation process (compiler + abstract machine) is described via a sequence of transformations all of which express an implementation choice. We characterize the essence of the Krivine machine and locate it in the design space of functional language implementations. We show that, even within the particular class of Krivine machines, hundreds of variants can be designed