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

Adaptive Detection of Instabilities: An Experimental Feasibility Study

We present an example of the practical implementation of a protocol for experimental bifurcation detection based on on-line identification and feedback control ideas. The idea is to couple the experiment with an on-line computer-assisted identification/feedback protocol so that the closed-loop system will converge to the open-loop bifurcation points. We demonstrate the applicability of this instability detection method by real-time, computer-assisted detection of period doubling bifurcations of an electronic circuit; the circuit implements an analog realization of the Roessler system. The method succeeds in locating the bifurcation points even in the presence of modest experimental uncertainties, noise and limited resolution. The results presented here include bifurcation detection experiments that rely on measurements of a single state variable and delay-based phase space reconstruction, as well as an example of tracing entire segments of a codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format (eps), need psfig macro to include them. Submitted to Physica

Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology

Mathematical modelling allows us to concisely describe fundamental principles in biology. Analysis of models can help to both explain known phenomena, and predict the existence of new, unseen behaviours. Model analysis is often a complex task, such that we have little choice but to approach the problem with computational methods. Numerical continuation is a computational method for analysing the dynamics of nonlinear models by algorithmically detecting bifurcations. Here we aim to promote the use of numerical continuation tools by providing an introduction to nonlinear dynamics and numerical bifurcation analysis. Many numerical continuation packages are available, covering a wide range of system classes; a review of these packages is provided, to help both new and experienced practitioners in choosing the appropriate software tools for their needs.Comment: 14 pages, 2 figures, 2 table