Singular Perturbation Research Articles

Rate-induced tipping can trigger plankton blooms

Plankton blooms are complex nonlinear phenomena whose occurrence can be described by the two-timescale (fast-slow) phytoplankton-zooplankton model introduced by Truscott and Brindley (Bulletin of Mathematical Biology 56(5):981–998, 1994). In their work, they observed that a sufficiently fast rise of the water temperature causes a critical transition from a low phytoplankton concentration to a single outburst: a so-called plankton bloom. However, the dynamical mechanism responsible for the observed transition has not been identified to the present day. Using techniques from geometric singular perturbation theory, we uncover the formerly overlooked rate-sensitive quasithreshold which is given by special trajectories called canards. The transition from low to high concentrations occurs when this rate-sensitive quasithreshold moves past the current state of the plankton system at some narrow critical range of warming rates. In this way, we identify rate-induced tipping as the underlying dynamical mechanism of largely unpredictable plankton blooms such as red tides, or more general, harmful algal blooms. Our findings explain the previously reported transitions to a single plankton bloom, and allow us to predict a new type of transition to a sequence of blooms for higher rates of warming. This could provide a possible mechanism of the observed increased frequency of harmful algal blooms.

The virtual element method with interior penalty for the fourth-order singular perturbation problem

We present the virtual element method with interior penalty to solve a fourth-order singular perturbation problem. In order to estimate the nonconformity error, the degrees of freedom on edges are changed to the moments of functions in the interior penalty scheme. To do this, we design a special H1-type projection that can be uniquely determined by the new degrees of freedom. With the help of the H1-type projection, the local discrete space is defined by imposing some restrictions on the local space of H2-conforming VE, such that it has the local H2 regularity. Then for the numerical method, we derive the error estimates under the condition of enough smooth solution and the uniform error estimates for cases with boundary layers. Finally, we display some numerical examples. From that, we see that the interior penalty method gives a better performance on the convergence than the theoretical predictions.

Dynamics of a Class of Prey–Predator Models with Singular Perturbation and Distributed Delay

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter [Formula: see text], the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

We study a class of two-dimensional nonlinear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a nontrivial adaptation of the standard properties of Schwartz symmetrization for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

Adaptive Active Disturbance Rejection Control for Uncertain Current Ripples Suppression of PMSM Drives

Under complex working conditions, there are uncertain periodic and aperiodic disturbances in the curr- ent loop of permanent magnet synchronous motor (PMSM) drives. These uncertain disturbances can lead to uncertain current ripples, deteriorating the current loop performance. To deal with the problem, this paper proposes an adaptive active disturbance rejection control (ADRC) for the current loop to suppress the uncertain current ripples. In the ada- ptive ADRC, the extended state observer is optimized by the adaptive resonant controller so that it can estimate both the uncertain periodic and aperiodic disturbances. Thus, the adaptive ADRC can suppress both the uncertain periodic and aperiodic disturbances to attenuate the uncer- tain current ripples in the current loop, and it does not need the disturbance information. In addition, the stability of the observer in the adaptive ADRC is systematically analyzed by the singular perturbation theory and the averaging theorem. Finally, the effectiveness of the adaptive ADRC is validated on a 2.2-kW PMSM platform.

Synchronization Control of Networked Two-Timescale Dynamic Agents: A Singular Perturbation Approach

Synchronization Control of Networked Two-Timescale Dynamic Agents: A Singular Perturbation Approach

How do various forces affect pressure waves in bubbly flows?

This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces acting on bubbles and studied their effects on pressure waves in a one-dimensional setting. Using a singular perturbation method, the Korteweg–de Vries–Burgers equation describing wave propagation was derived. The following results were obtained: (i) Bjerknes force on the bubbles enhanced the nonlinearity, dissipation, and dispersion of the waves; (ii) Drag, gravity, and buoyancy forces acting on the bubbles increased wave dissipation; (iii) Thermal conduction had the most substantial dissipation effect, followed by acoustic radiation, drag, buoyancy, and gravity. We confirmed that the dissipation due to forces on gas bubbles was quantitatively minor.

Transient nucleation driven by solvent evaporation.

We theoretically investigate homogeneous crystal nucleation in a solution containing a solute and a volatile solvent. The solvent evaporates from the solution, thereby continuously increasing the concentration of the solute. We view it as an idealized model for the far-out-of-equilibrium conditions present during the liquid-state manufacturing of organic electronic devices. Our model is based on classical nucleation theory, taking the solvent to be a source of the transient conditions in which the solute drops out of the solution. Other than that, the solvent is not directly involved in the nucleation process itself. We approximately solve the kinetic master equations using a combination of Laplace transforms and singular perturbation theory, providing an analytical expression for the nucleation flux. Our results predict that (i) the nucleation flux lags slightly behind a commonly used quasi-steady-state approximation. This effect is governed by two counteracting effects originating from solvent evaporation: while a faster evaporation rate results in an increasingly larger influence of the lag time on the nucleation flux, this lag time itself is found to decrease with increasing evaporation rate. Moreover, we find that (ii) the nucleation flux and the quasi-steady-state nucleation flux are never identical, except trivially in the stationary limit, and (iii) the initial induction period of the nucleation flux, which we characterize as a generalized induction time, decreases weakly with the evaporation rate. This indicates that the relevant time scale for nucleation also decreases with an increasing evaporation rate. Our analytical theory compares favorably with results from a numerical evaluation of the governing kinetic equations.

Transient synchronization stability of photovoltaics integration by singular perturbation analysis

The integration of large-scale photovoltaics (PVs) into the power grid has significantly altered the transient synchronization dynamics of traditional power systems dominated by synchronous generators (SGs) and posed great challenges to modeling and analysis of PVs integration. In this paper, the transient synchronization stability of the PV-SG system is studied using the singular perturbation technique. Firstly, a nonlinear model of a PV-SG system is established to reveal the multiscale transient synchronization characteristics. Further, the full system is decomposed into a slow subsystem and a fast subsystem by the singular perturbation technique. The fast subsystem containing the dynamics of the DC voltage control, terminal voltage control, and phase-locked loop, and the slow subsystem containing the dynamics of rotor motion can perfectly reflect the dynamics of the full system within the electromagnetic and electromechanical timescales, respectively. The proposed model provides a clearer physical picture of dynamics in the PV-SG system within the electromagnetic and electromechanical timescales. Subsequently, the stability of the slow and fast subsystems is investigated using the energy function and eigenvalue analysis methods, respectively. Meanwhile, the impacts of various operating, control, and structural parameters on the transient synchronization stability are uncovered. Different from the most existing research endeavors on the wide simulations of the PVs integration, the impact of PVs on the synchronization dynamics of SGs without considering the dynamical characteristics of the PV system, and the transient synchronization stability analyses of the PLL-based voltage source converter systems, it is the key contribution to study the transient synchronization dynamical characteristics of the PV system and its interaction with the SG under different timescales. All these are helpful and easy to extend to more complicated PV-SG systems. Finally, the analysis results are validated by extensive simulations.

Distributed Formation Control for Multiple Quadrotor UAVs Based on Distributed Estimator and Singular Perturbation System

Distributed Formation Control for Multiple Quadrotor UAVs Based on Distributed Estimator and Singular Perturbation System

Data-Driven and Low-Rank Implementations of Balanced Singular Perturbation Approximation

Data-Driven and Low-Rank Implementations of Balanced Singular Perturbation Approximation

Numerical analysis of singularly perturbed parabolic reaction diffusion differential difference equations

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of ‘Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions’ (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings.

Hybrid Adjusting Variables-Dependent Event-Based Finite-Time State Estimation for Two-Time-Scale Markov Jump Complex Networks.

This article investigates the problem of dynamic event-triggered finite-time H∞ state estimation for a class of discrete-time nonlinear two-time-scale Markov jump complex networks. A hybrid adjusting variables-dependent dynamic event-triggered mechanism (DETM) is proposed to regulate the releases of measurement outputs of a node to a remote state estimator. Such a DETM contains both an additive dynamically adjusting variable (DAV) and a multiplicative adaptively adjusting variable. The aim is to design a DETM-based mode-dependent state estimator, which guarantees that the resultant error dynamics is stochastically finite-time bounded with H∞ performance. By constructing a mode-dependent Lyapunov function with multiple DAVs and a singular perturbation parameter associated with time scales, a matrix-inequalities-based sufficient condition is derived, the feasible solutions of which facilitate the design of the parameters of the state estimator. The validity of the designed state estimator and the superiority of the devised DETM are verified by two examples. It is verified that the devised DETM is capable of saving network resources and simultaneously improving the estimation performance.

On the balanced truncation error bound and sign parameters from arrowhead realizations

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.

OME[formula omitted]/air mixtures: The structure and propagation mechanism of their laminar premixed flames

Oxymethylene dimethyl ethers (CH3O(CH2O)xCH3 or OMEx) are synthetic substitutes of diesel and jet fuel. They can achieve soot-less combustion and have been shown to exhibit favourable ignition characteristics. With the aim to explore their combustion physics, the structure of their flames is analysed using the algorithmic tools of the Computational Singular Perturbation (CSP) method. It is demonstrated that the flame structure and the mechanism for its propagation is encapsulated in the fastest explosive mode, introduced by chemical dynamics in a narrow region of the flame front. This narrow region extends around the point where the maximum heat release rate (MHRR) is recorded. CSP diagnostics identify (i) reactions H + O2→ O + OH and CO + OH → CO2 + H beyond MHRR as initiating an upstream diffusion of heat and H radicals, both of which are generated by the exothermic reaction H2 + OH → H2O + H, (ii) a downstream convective motion that ensues and (iii) chemical activity that is initiated ahead of MHRR and sustained by the convective transport of fuel and oxygen. CSP also identifies differences in the action of the upstream diffusion of heat and H radicals and of the downstream transport of fuel and oxidiser. It is also shown that the fastest explosive mode far from the neighbourhood of MHRR has no influence on the structure and propagation of the flame front. Although OME2−4/air mixtures exhibit drastically different dynamics in the context of homogeneous autoignition, very small differences are recorded in terms of flame structure and propagation. The findings suggests that, since practically all major processes driving the flame front propagation are similar to those of hydrocarbons, OMEx fuels can be used as “drop-in” fuels in currently employed burners.

Uniform estimates for conforming Galerkin method for anisotropic singularly perturbed elliptic problems

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools.

Family of Skeletal Reaction Mechanisms for Methane–Oxygen Combustion in Rocket Propulsion

Analyzing methane–oxygen rocket propellant combinations requires suitable modeling of the major chemical reaction processes. Although several detailed kinetic mechanisms for methane oxidation in air exist, most do not reproduce the reaction pathways of high-pressure methane–oxygen combustion, typical of liquid rocket engines. Moreover, when large-scale computational fluid dynamics simulations are pursued, detailed reaction schemes are not computationally viable. In the present study, we identify a reliable detailed kinetic scheme for liquid rocket applications, and then we perform a wide reduction campaign leveraging computational singular perturbation theory. Enforcing various reduction targets, we obtain a family of seven skeletal schemes, including 11–39 species. Each mechanism targets different combustion modes, namely, homogeneous ignition, complex flows and flame extinction, premixed burning, reaction processes under intense turbulent mixing, and largely off-stoichiometric mixtures, typical of rocket engine preburners. We test the skeletal mechanisms against meaningful validation targets, attaining appreciable predictive accuracy compared with the detailed parent scheme. We expect the proposed family of skeletal schemes to offer a wide and flexible range of solutions—in terms of size, accuracy, and dominant combustion mode—for performing large-scale yet cost-affordable computational fluid dynamics of methane–oxygen flames under rocket-engine-relevant conditions.

Analytical and Computational Approaches for Bi-Stable Reaction and p-Laplacian Diffusion Flame Dynamics in Porous Media

In this paper, we present a mathematical approach for studying the changes in pressure and temperature variables in flames. This conception extends beyond the traditional second-order Laplacian diffusion model by considering the p-Laplacian operator and a bi-stable reaction term, thereby providing a more generalized framework for flame diffusion analysis. Given the structure of our equations, we provide the boundedness and uniqueness of the solutions in a weak sense from both analytical and numerical approaches. We further reformulate the governing equations in the context of traveling wave solutions, applying singular geometric perturbation theory to derive the analytical expressions of these profiles. This theoretical development is complemented by numerical assessments, which not only validate our theoretical predictions, but also optimize the traveling wave speed to minimize the error between numerical and analytical solutions. Additionally, we explore self-similar structured solutions. The paper then concludes with a perspective on future research, with emphasis being placed on the need for experimental validation in laboratory settings. Such empirical studies could test the robustness of our model and allow for refinement based on actual measurements, thereby broadening the applicability and accuracy of our findings in practical scenarios.

Supercloseness of the LDG method for a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh

In this study, the focus is on exploring the supercloseness property of the local discontinuous Galerkin (LDG) method in the context of a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh. By developing specialized local Gauss–Radau projections in the two-dimensional case, and establishing a novel interpolation premised on the special projections, supercloseness of an optimal order k+1 can be achieved on Bakhvalov-type mesh. Notably, the obtained result remains unaffected by the singular perturbation parameter ɛ, indicating that the limitations encountered in previous studies, where the supercloseness results on Bakhvalov-type mesh were always impacted by the logarithmic factor ln1ɛ are overcome.

Asymptotic study of critical wave fronts for parameter-dependent Born–Infeld models: physically predicted behaviors and new phenomena

In this paper, we study some parameter-dependent reaction-diffusion models governed by the Born–Infeld (or Minkowski) operator. In dependence on two parameters , related to the field strength and to the diffusivity, we investigate the limit critical speed for traveling fronts, together with the limit behavior of the associated critical profiles. As the two main results, on the one hand we rigorously show that for arbitrarily large electric fields the critical speed and the critical profile converge to the ones of the linear diffusion problem, agreeing with the well-known physical prediction that, in this case, Born’s law and Maxwell’s law coincide. Such a result is accompanied by a counterpart for arbitrarily small electric fields and a complete analysis for vanishing/large diffusion. On the other hand, we prove the onset of a new and unexpected phenomenon for the singular perturbation problem: the critical speed of propagation does not converge to zero and, for KPP or combustion-type reactions, the limit front profile becomes sharp on one side only. In fact, this latter coincides with the C 1-gluing of a piecewise linear profile having slope equal to 1 when nonconstant, and of a regular inviscid profile propagating at the limit speed. The gluing point and the value of the limit speed are determined explicitly. The outcomes of the whole discussion, based on a careful analysis of the first-order reduction associated with the original equation, show substantial differences and peculiarities of the Born–Infeld operator with respect to linear or saturating diffusions, and are supported by several numerical experiments.