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We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay., Comment: postprint version

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006)., Comment: minor revision, correcting mistakes in formulation of Lemma A.3 (which are also present in the Journal paper): center $x$ of neighborhood must be in $C^1$, which is the case for the main theorem. Further clarification in proof of Lemma A.3

It is often known, from modelling studies, that a certain mode of climate tipping (of the oceanic thermohaline circulation, for example) is governed by an underlying fold bifurcation. For such a case we present a scheme of analysis that determines the best stochastic fit to the existing data. This provides the evolution rate of the effective control parameter, the variation of the stability coefficient, the path itself and its tipping point. By assessing the actual effective level of noise in the available time series, we are then able to make probability estimates of the time of tipping. This new technique is applied, first, to the output of a computer simulation for the end of greenhouse Earth about 34 million years ago when the climate tipped from a tropical state into an icehouse state with ice caps. Second, we use the algorithms to give probabilistic tipping estimates for the end of the most recent glaciation of the Earth using actual archaeological ice-core data., Comment: minor corrections

Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction for characteristic matrices for systems of linear delay-differential equations with periodic coefficients. First, we show that matrices constructed in this way can have a discrete set of poles in the complex plane, which may possibly obstruct their use when determining the stability of the linear system. Then we modify and generalize the original construction such that the poles get pushed into a small neighborhood of the origin of the complex plane., Comment: 17 pages, 1 figure

If one wants to explore the properties of a dynamical system systematically one has to be able to track equilibria and periodic orbits regardless of their stability. If the dynamical system is a controllable experiment then one approach is a combination of classical feedback control and Newton iterations. Mechanical experiments on a parametrically excited pendulum have recently shown the practical feasibility of a simplified version of this algorithm: a combination of time-delayed feedback control (as proposed by Pyragas) and a Newton iteration on a low-dimensional system of equations. We show that both parts of the algorithm are uniformly stable near the saddle-node bifurcation: the experiment with time-delayed feedback control has uniformly stable periodic orbits, and the two-dimensional nonlinear system which has to be solved to make the control non-invasive has a well-conditioned Jacobian., Comment: 6 pages, 3 figures

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos., Comment: 26 pages, 9 figures (corrected figure caption in Fig C2)