Mitigation of tipping point transitions by time-delay feedback control

Control of ecological networks: Abundance control or ecological regulation?

Complex ecosystems often exhibit a tipping point around which a small perturbation can lead to the loss of the basic functionality of ecosystems. It is challenging to develop a control strategy to bring ecosystems to the desired stable states. Typically, two methods are employed to restore the functionality of ecosystems: abundance control and ecological regulation. Abundance control involves directly managing species abundance through methods such as trapping, shooting, or poisoning. On the other hand, ecological regulation is a strategy for ecosystems to self-regulate through environment improvement. To enhance the effectiveness of ecosystem recovery, we propose adaptive regulation by combining the two control strategies from mathematical and network science perspectives. Criteria for controlling ecosystems to reach equilibrium with or without noise perturbation are established. The time and energy costs of restoring an ecosystem to equilibrium often determine the choice of control strategy, thus, we estimate the control costs. Furthermore, we observe that the regulation parameter in adaptive regulation affects both time and energy costs, with a trade-off existing between them. By optimizing the regulation parameter based on a performance index with fixed weights for time and energy costs, we can minimize the total cost. Moreover, we discuss the impact of the complexity of ecological networks on control costs, where the more complex the networks, the higher the costs. We provide corresponding theoretical analyses for random networks, predator-prey networks, and mixture networks.

Preventing a global transition to thermoacoustic instability by targeting local dynamics

The burning of fossil fuels to generate power produces harmful emissions. Lowering such emissions in gas turbine engines is possible by operating them at fuel-lean conditions. However, such strategies often fail because, under fuel-lean conditions, the combustors are prone to catastrophic high-amplitude oscillations known as thermoacoustic instability. We reveal that, as an operating parameter is varied in time, the transition to thermoacoustic instability is initiated at specific spatial regions before it is observed in larger regions of the combustor. We use two indicators to discover such inceptive regions: the growth of variance of fluctuations in spatially resolved heat release rate and its spatiotemporal evolution. In this study, we report experimental evidence of suppression of the global transition to thermoacoustic instability through targeted modification of local dynamics at the inceptive regions. We strategically arrange slots on the flame anchor, which, in turn, reduce the local heat release rate fluctuations at the inceptive regions and thus suppress the global transition to thermoacoustic instability. Our results open new perspectives for combustors that are more environmental-friendly.

Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model

We study the mitigation of climate tipping point transitions using an energy balance model. The evolution of the global mean surface temperature is coupled with the CO 2 concentration through the green-house effect. We model the CO 2 concentration with a stochastic delay differential equation (SDDE), accounting for various carbon emission and capture scenarios. The resulting coupled system of SDDEs exhibits a tipping point phenomena: if CO 2 concentration exceeds a critical threshold (around 478   ppm ), the temperature experiences an abrupt increase of about six degrees Celsius. We show that the CO 2 concentration exhibits a transient growth which may cause a climate tipping point, even if the concentration decays asymptotically. We derive a rigorous upper bound for the CO 2 evolution which quantifies its transient and asymptotic growths, and provides sufficient conditions for evading the climate tipping point. Combining this upper bound with Monte Carlo simulations of the stochastic climate model, we investigate the emission reduction and carbon capture scenarios that would avert the tipping point.

Mitigation of rare events in multistable systems driven by correlated noise

We consider rare transitions induced by colored noise excitation in multistable systems. We show that undesirable transitions can be mitigated by a simple time-delay feedback control if the control parameters are judiciously chosen. We devise a parsimonious method for selecting the optimal control parameters, without requiring any Monte Carlo simulations of the system. This method relies on a new nonlinear Fokker-Planck equation whose stationary response distribution is approximated by a rapidly convergent iterative algorithm. In addition, our framework allows us to accurately predict, and subsequently suppress, the modal drift and tail inflation in the controlled stationary distribution. We demonstrate the efficacy of our method on two examples, including an optical laser model perturbed by multiplicative colored noise.

Data-driven prediction of multistable systems from sparse measurements

We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) it is compatible with the precomputed library and (ii) it is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to the scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction-diffusion equation, arising in pattern formation, which has four asymptotically stable steady states, and a FitzHugh-Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction-diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95% accuracy when a moderate number of labeled data are used. For the FitzHugh-Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90% accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions.