Some Applications of Bifurcation Formulae to the Period Maps of Delay Differential Equations

Our purpose is to present some applications of the bifurcation formulae derived in [13] for periodic delay differential equations. We prove that a sequence of Neimark-Sacker bifurcations occurs as the parameter increases. For some special classes of equations, easily checkable conditions are given to determine the direction of the bifurcation of the time-one map

Mapping dynamical systems with distributed time delays to sets of ordinary differential equations

Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. It is shown that an exact mapping onto a set of $N+1$ ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents. The number of auxiliary variables one needs to introduce, $N$, is inversely proportional to the variance of the delay distribution. The case of a single delay is therefore recovered when $N\to\infty$. Using this approach, denoted the kernel series framework, we examine systematically how the bifurcation phase diagram of the Mackey-Glass system changes under the influence of distributed delays. We find that local properties, f.i. the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels. Period-doubling transitions and the onset of chaos, which involve non-local properties of the flow, are found in contrast to be more sensible to distributed delays. Our results indicate that modeling approaches of real-world processes should take the effects of distributed delay times into account.Comment: 16 pages, 6 figure

Role of delay for the symmetry in the dynamics of networks

PACS number(s): 05.45.Xt, 89.75.Kd, 89.75.Hc, 02.30.KsThe symmetry in a network of oscillators determines the spatiotemporal patterns of activity that can emerge. We study how a delay in the coupling affects symmetry-breaking and -restoring bifurcations. We are able to draw general conclusions in the limit of long delays. For one class of networks we derive a criterion that predicts that delays have a symmetrizing effect. Moreover, we demonstrate that for any network admitting a steady-state solution, a long delay can solely advance the first bifurcation point as compared to the instantaneous-coupling regime.This work was partially supported by the Interuniversity Attraction Poles program Photonics@be of the Belgian Science Policy Office under Grant No. IAP VI-10 by MICINN (Spain) under Project No. DeCoDicA (TEC2009-14101) and by the project PHOCUS (EU FET-Open Grant No. 240763). S. Yanchuk and P. Perlikowski are gratefully acknowledged for fruitful discussions.Peer reviewe