728 research outputs found

### 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 paperās 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 Sieberās research has received funding from the European Unionās 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

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