Tipping points near a delayed saddle node bifurcation with periodic forcing

We consider the effect on tipping from an additive periodic forcing in a canonical model with a saddle node bifurcation and a slowly varying bifurcation parameter. Here tipping refers to the dramatic change in dynamical behavior characterized by a rapid transition away from a previously attracting state. In the absence of the periodic forcing, it is well-known that a slowly varying bifurcation parameter produces a delay in this transition, beyond the bifurcation point for the static case. Using a multiple scales analysis, we consider the effect of amplitude and frequency of the periodic forcing relative to the drifting rate of the slowly varying bifurcation parameter. We show that a high frequency oscillation drives an earlier tipping when the bifurcation parameter varies more slowly, with the advance of the tipping point proportional to the square of the ratio of amplitude to frequency. In the low frequency case the position of the tipping point is affected by the frequency, amplitude and phase of the oscillation. The results are based on an analysis of the local concavity of the trajectory, used for low frequencies both of the same order as the drifting rate of the bifurcation parameter and for low frequencies larger than the drifting rate. The tipping point location is advanced with increased amplitude of the periodic forcing, with critical amplitudes where there are jumps in the location, yielding significant advances in the tipping point. We demonstrate the analysis for two applications with saddle node-type bifurcations

Automatic Detection of Egg Shell Cracks

The challenge was to find a reliable, non-intrusive means of detecting cracks in eggs. Intensity data from eggs were collected by VisionSmart for the group to analyse. Given the short time period three main questions were addressed. 1) Is there a feature of the intensity data which detects, and discriminates between pinholes, cage marks and cracks? 2) Are there ways to improve the current data collection process? 3) Are there other data collection methods which should be tried? A partial positive response to 1) is presented and describes the many problems that arose. Some answers to 2) and 3) are also presented

Dynamic tipping in the non-smooth Stommel-box model, with fast oscillatory forcing

We study the behavior at tipping points close to non-smooth fold bifurcations in non-autonomous systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modeling thermohaline circulation. We obtain explicit asymptotic expressions for the behavior at tipping points in the settings of both slowly varying freshwater forcing and rapidly oscillatory fluctuations. The results, based on combined multiple scale and local analyses, provide conditions for the sudden transitions between temperature-dominated and salinity-dominated states. In the context of high frequency oscillations, a multiple scale averaging approach can be used instead of the usual geometric approach normally required for piecewise-smooth continuous systems. The explicit parametric dependencies of advances and lags in the tipping show a competition between dynamic features of the model. We make a contrast between the behavior of tipping points close to both smooth Saddle–Node Bifurcations and the non-smooth systems studied on this paper. In particular we show that the non-smooth case has earlier and more abrupt transitions. This result has clear implications for the design of early warning signals for tipping in the case of the non-smooth dynamical systems which often arise in climate models.</p

Qualitative changes in bifurcation structure for soft vs hard impact models of a vibro-impact energy harvester

Funding Information: The authors gratefully acknowledge partial funding for this work from NSF-CMMI (No. 2009270) and EPSRC (No. EP/V034391/1).Peer reviewedPublisher PD

WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming

Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise. For a robust recovery against noise and the choice of hyper-parameters, we introduce two new mechanisms, narrow-fit and trimming, for the coefficient support and value recovery, respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming of the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to $100\%$ noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations

Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude, epsilson [e]. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms, or, if there exists an attracting period-n solution when e = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1/n, for sufficiently small e > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to the square-root of e, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.Comment: Submitted to: SIAM J. Appl. Dyn. Sy

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and reset and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.Comment: 22 page