Probability of noise- and rate-induced tipping

We propose an approximation for the probability of tipping when the speed of parameter change and additive white noise interact to cause tipping. Our approximation is valid for small to moderate drift speeds and helps to estimate the probability of false positives and false negatives in early-warning indicators in the case of rate- and noise-induced tipping. We illustrate our approximation on a prototypical model for rate-induced tipping with additive noise using Monte-Carlo simulations. The formula can be extended to close encounters of rate-induced tipping and is otherwise applicable to other forms of tipping. We also provide an asymptotic formula for the critical ramp speed of the parameter in the absence of noise for a general class of systems undergoing rate-induced tipping.Comment: 14 pages, 9 figures https://link.aps.org/doi/10.1103/PhysRevE.95.05220

Systematic experimental exploration of bifurcations with non-invasive control

We present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular non-invasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced periodic orbits in experiments. To demonstrate the use of our non-invasive control, we trace out experimentally the resonance surface of a periodically forced mechanical nonlinear oscillator near the onset of instability, around two saddle-node bifurcations (folds) and a cusp bifurcation.Comment: revised and extended version (8 pages, 7 figures

Evolution of the L1 halo family in the radial solar sail CRTBP

We present a detailed investigation of the dramatic changes that occur in the $\mathcal{L}_1$ halo family when radiation pressure is introduced into the Sun-Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. 1991) the families can be fully mapped out as the parameter $\beta$ is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at $\beta\approx0.0387$ acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at $\beta\approx0.289$, resulting in a third new family. The linear stability of the families changes rapidly at low values of $\beta$, with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter $\beta$ and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of $\beta$ there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.Comment: 31 pages, 17 figures, accepted by Celestial Mechanics and Dynamical Astronom

Inverse-square law between time and amplitude for crossing tipping thresholds

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses} near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher dimensional system, a model for the Indian Summer Monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.Comment: 20 pages, 6 figures, 1 table, Supplementary Material found at https://figshare.com/articles/Monsoon_supplementary_material_final_pdf/760582

Generic stabilisability for time-delayed feedback control

Time delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that there are no restrictions in principle. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilise this orbit with extended time-delayed feedback control. While the paper’s techniques are based on linear stability analysis, they exploit the specific properties of linearisations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.Jan Sieber’s research has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement number 643073

Characteristic matrices for linear periodic delay differential equations

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 figur

Nonlinear dynamic Interactions between flow-induced galloping and shell-like buckling

Acknowledgement The research of J.S. is supported by EPSRC Grant EP/J010820/1.Peer reviewedPublisher PD