Nonlinear Stability Conditions Research Articles

Global stability of Oldroyd-B fluids in plane Couette flow

We prove conditions for global nonlinear stability of Oldroyd-B viscoelastic fluid flows in the Couette shear flow geometry. Global stability is inferred by analysing a new functional, called a perturbation entropy, to quantify the magnitude of the polymer perturbations from their steady-state values. The conditions for global stability extend, in a physically natural manner, classical results on global stability of Newtonian Couette flow.

A High Robust Optimal Nonlinear Control with MPPT Speed for Wind Energy Conversion System (WECS) Based on Doubly Fed Induction Generator (DFIG)

As part of efforts to improve wind energy production using a doubly fed induction generator (DFIG), this paper presents modeling with power maximization through control of speed (MPPT) and control of stator active and reactive power for doubly fed induction generator. The quality of the generated energy and the performance of wind energy conversion systems (WECS) based on DFIG are affected by uncertainties and external disturbances. Therefore, the system requires high-performance control under the nonlinearity influence and the applied external disturbances. This work provides a new contribution and approach that combines nonlinear control (Backstepping) and optimal control (LQR). Backstepping control has been applied to give a good performance using nonlinear strategy and stability condition, but its main drawback is that it is relatively robust for sudden changes in wind speed or any external disturbances as well as it is sensitive against the uncertainties, which may lead to a chattering and an instability as well as overshoot or undershoot. To overcome the problems mentioned above the optimal control (LQR) has been applied to address this drawback in nonlinear control. Furthermore, it, the hybrid approach of Backstepping and LQR control resulting a robust control under the hardest scenarios during the wind system operating as well as provides a good performance in terms of chattering, stability and other performance characteristics which help to produce a high-quality energy. Overall, integrating LQR control with Backstepping can lead to a more powerful, flexible, and efficient control system with improved performance and reduced computational complexity.

Dynamical property and triadic interaction of Beltrami-type rotating waves

In this work, on the one hand, we continue with the study of the dynamic property of Beltrami flows [R. González, “Dynamics of non-axisymmetric Beltrami flows,” Phys. Fluids 26, 114104 (2014)], extending its scopes to non-stationary flows in the rotating system, which allowed us to classify the rotating waves on the basis of their phase velocities. On the other hand, and in accordance with this classification, we study the resonant triadic interaction of these waves. For this purpose, we use the expansion in a Chandrasekhar–Kendall basis for an infinite tube, in an analogous procedure to the one carried out by Waleffe [“The nature of triad interactions in homogeneous turbulence,” Phys. Fluids A 4, 350–363 (1992)], to study the interaction of plane Beltrami waves. Taking an equilibrium of the resulting non-linear equations for the amplitudes of the waves, we consider their linear and non-linear stability. Regarding their linear stability, we see that unlike plane Beltrami waves, their stability depends not only on the relative helicities but also on an interaction factor that depends on the properties and the co-rotating or counter-rotating character of the interacting waves. On the other hand, for non-linear stability dependent on the same parameters as those of linear stability, we find, for one case of analysis, that there is non-linear instability only for some interactions of two co-rotating waves with a counter-rotating one and we exhibit criteria that are sufficient conditions of non-linear stability or non-linear instability.

Effects of uniform and periodic magnetic fields at the nonlinear stability of three magnetic fluids in porous media

The current manuscript tackles the interaction between three viscous magnetic fluids placed on three layers and saturated in porous media. Two of them fill half of the spaces above and below a thin layer that lies in the middle region. All layers are laterally extended to infinity in both horizontal directions. All fluids move in the same horizontal direction with different uniform velocities and are driven by pressure gradients. The system is stressed by tangential stationary/periodic magnetic fields. The viscous potential theory is used to simplify the mathematical procedure. The motion of the fluids is described by the Brinkman–Darcy equations, and Maxwell equations are used for the magnetic field. The work has been motivated in light of the engineering applications like petroleum products manufacturing and the electromagnetic field effect that can be used to control the growth of the perturbation and then the recovery of crude oil from the pores of reservoir rocks. The nonlinear technique typically relies on solving linear equations of motion and presenting the nonlinear boundary conditions. The novelty of the problem concerns the nonlinear stability of the double interface under the impact of periodic magnetic fields. Therefore, the approach has resulted in two nonlinear characteristic differential equations governing the surface displacements. Accordingly, the development amplitudes of surface waves are designated by two nonlinear Schrödinger equations. Stability is theoretically analyzed; the nonlinear stability criteria are derived, and the corresponding nonlinear stability conditions are explored in detail. The analysis reveals the resonance as well as the non-resonance situations. Approximate bounded solutions of the perturbed interfaces are estimated. It is found that the thickness of the intermediate layer is represented as a function of time and plotted. The impact of different parameters on the stability profile is investigated. For the middle layer, it is observed that magnetic permeability and viscosity have a stabilizing effect. By contrast, basic streaming and permeability have a destabilizing influence. Due to different saturation amplitude at each case, the analysis of the periodic case shows that the lower interface is much more stable than the upper one.

Investigation of dynamical stability of vehicle lateral motion control system with feedback delay

<p indent="0mm">Time delay is one of the key factors damaging the driving safety in a vehicle lateral motion control system. This paper establishes a nonlinear vehicle model considering yaw and lateral motion, and two types of nonlinear steering characteristics and motion stability conditions are analyzed using handling diagrams. Next, a PD-based active yaw moment controller with time delay is applied to construct a complete hybrid vehicle system with delayed control. Firstly, the effect of time delay on the general stability and stability boundary of the vehicle system is studied using the location of the characteristic roots and the frequency domain method, wherein the results indicate that the lateral stability of the vehicle is significantly weakened under time delay, and the instability boundary can be reached under a substantial time delay. Furthermore, the optimization factor characterizing the system decay rate is introduced, and the optimization method for the control parameter of the vehicle lateral system is designed. The stability region is optimally divided to select the optimal control gain set under some delay. The optimization is compared and verified based on the steady-state simulation and dynamic test, and the results validate its effectiveness in improving the lateral stability and suppressing the effect of time delay, wherein the optimized dynamic responses of the system follow the target trajectory. Based on the relations between the optimization factor and vehicle characteristics, the critical maximum of the optimization factor is proposed as the quantitative index to evaluate the vehicle lateral stability under time delay, where its general parametric expressions are derived by exploring two types of critical instability conditions. Finally, the effectiveness and efficiency of the index in quantifying the delayed system stability and predicting the time-delay boundary of instability are sufficiently verified in steady-state and dynamic simulation.

Nonlinear Modeling and Global Stability Condition of Single-Phase Grid-Tied Inverter Considering SRF-PLL and Duty-Cycle Saturation

Considering that the traditional linearization method cannot accurately analyze the grid-tied inverter (GTI) system with nonlinear nature, in the single-phase case, a novel nonlinear model for the GTI system considering nonlinear factors, including synchronous reference frame phase-locked loop (PLL) and duty-cycle saturation in pulsewidth modulation, is proposed, and the corresponding global asymptotic stability condition is given. Specifically, some system nonlinear factors, e.g., duty-cycle saturation phenomenon and Park ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> ) transformation and cosine operation in second-order generalized integrator-based PLL (SOGI-PLL), are investigated, and their sector-bounded conditions are derived. In order to seek less conservative results, the system state of the SOGI-PLL is restructured such that the linear part of the PLL system can be stabilized. Then, the nonlinear state-space model with sector bounded for the whole GTI system is formulated. Furthermore, the deduced sector conditions are introduced into the derivation of the Lyapunov function by using the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S-procedure</i> lemma, and the global asymptotic stability condition of the inverter system is, thus, obtained. The system stability regions under different grid impedances, current references, and control parameters are further analyzed. Finally, the effectiveness of the proposed theories is fully supported by transient experimental results.

Impact of early treatment programs on Swine flu infection with optimal controls: Mathematical model

This manuscript focuses on the impact of early treatment programs on swine flu disease transmission among the population. In this manuscript, a nonlinear Susceptible-Exposed-Infected-Recovered (SEIR) model with early Treatment programs are developed to examine the transmission dynamics of Swine flu infection with the help of the system of ordinary differential equations. The characteristics of the model are investigated by the basic reproduction number. We analyzed that the model exhibits using stability theory of differential equations, the disease-free equilibrium is linearly stable for R01. Also, conditions for non-linear stability are derived. Sensitivity indices for basic reproduction and also optimal control measures for swine flu are obtained. Further, numerical simulation for the model is supported by relevant graphs.

Dual-level toxicity assessment of biodegradable pesticides to aquatic species

The use of ecological models to assess the effects of pesticides on the structure and function of aquatic ecosystems is of growing interest. Of utmost concern is assessment of pesticides that have the potential to biodegrade into metabolites that are as toxic as the parent pesticide. In this work, a mathematical model to predict and evaluate the effect of two pesticides on the population of aquatic biospecies where both pesticides are bio-available in water and sediments with one of the pesticides capable of biodegrading into the other but not vice versa is formulated and analyzed. Conditions for nonlinear stability, instability and Hopf-bifurcation are obtained. The model undergoes a Hopf-bifurcation when the rate of discharge of pesticides crosses a critical value, so that the population of the aquatic species follows an oscillatory pattern. Four hypotheses involving the concentration of the pesticides and the population of the aquatic species where qualitatively investigated: the aquatic species will completely die out with time, the population of the aquatic biospecies will remain under certain conditions, the pesticide will continually remain in water and there will be a periodic variation in the population of the aquatic species over time. Results also indicate that the biodegradation potential of one of the pesticides had significant effect on the population dynamics of the aquatic species.

Stability analysis of PD AQM control for delay models of TCP networks

ABSTRACT This paper focuses on the local stability analysis of nonlinear delay models of transmission control protocol/active queue management (TCP/AQM) networks by using a Proportional-Derivative (PD) controller as AQM strategy. More precisely, we derive the complete set of PD controllers that exponentially stabilises the corresponding linear delay system. Nonlinear sufficient stability conditions for the local asymptotic stability of the equilibrium are derived by means of the Lyapunov–Krasovskii functional approach. The robustness issue to uncertainty in the network parameters is also addressed. The results are illustrated by means of some numerical examples.

Non-linear CFL Conditions Issued from the von Neumann Stability Analysis for the Transport Equation

This paper presents a theory of the possible non-linear stability conditions encountered in the simulation of convection dominated problems. Its main objective is to study and justify original CFL-like stability conditions thanks to the von Neumann stability analysis. In particular, we exhibit a wide variety of stability conditions of the type $$\Delta t\le C \Delta x^\alpha $$ with $$\Delta t$$ the time step, $$\Delta x$$ the space step, and $$\alpha $$ a rational number within the interval [1, 2]. Numerical experiments corroborate these theoretical results.

Linearity of stability conditions

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.

Driver’s Anticipation and Memory Driving Car-Following Model

We developed a new car-following model to investigate the effects of driver anticipation and driver memory on traffic flow. The changes of headway, relative velocity, and driver memory to the vehicle in front are introduced as factors of driver’s anticipation behavior. Linear and nonlinear stability analyses are both applied to study the linear and nonlinear stability conditions of the new model. Through nonlinear analysis a modified Korteweg-de Vries (mKdV) equation was constructed to describe traffic flow near the traffic near the critical point. Numerical simulation shows that the stability of traffic flow can be effectively enhanced by the effect of driver anticipation and memory. The starting and breaking process of vehicles passing through the signalized intersection considering anticipation and driver memory are presented. All results demonstrate that the AMD model exhibit a greater stability as compared to existing car-following models.

Stability of the Malvinas Current.

Deterministic and probabilistic tools from nonlinear dynamics are used to assess enduring near-surface Lagrangian aspects of the Malvinas Current. The deterministic tools are applied to a multiyear record of velocities derived from satellite altimetry data, revealing a resilient cross-stream transport barrier. This is composed of shearless-parabolic Lagrangian coherent structures (LCSs), which, extracted over sliding time windows along the multiyear altimetry-derived velocity record, lie in near-coincidental position. The probabilistic tools are applied on a large collection of historical satellite-tracked drifter trajectories, revealing weakly communicating flow regions as basins of attraction for long-time asymptotic almost-invariant sets on either side of the altimetry-derived barrier. Shearless-parabolic LCSs are detected for the first time from altimetry data, and their significance is supported on satellite-derived ocean color data, which reveal shapes that quite closely resemble the peculiar V shapes, dubbed "chevrons," that have recently confirmed the presence of similar LCSs in the atmosphere of Jupiter. Finally, using available in situ velocity and hydrographic data, sufficient and necessary conditions for nonlinear symmetric stability are found to be satisfied, suggesting a duality between Lagrangian and Eulerian stability for the Malvinas Current.

Distributed PI Control For Heterogeneous Nonlinear Platoon of Autonomous Connected Vehicles

Abstract In this paper we present a distributed PI-based control law that guarantees the platoon formation to track leader velocity with a pre-fixed inter-vehicular distance, when vehicles are heterogeneous in both parameters and nonlinear drivetrain dynamics. Moreover, differently from most of the approaches in the literature, the proposed PI protocol intrinsically compensates for the nonlinear, heterogeneous and uncertain drivetrain dynamics without requiring any feedforward control action. We formulate sufficient conditions for closed-loop heterogeneous non-linear vehicular network stability that can be used to tune the PI control parameters. Simulation results confirm the effectiveness of the proposed PI controller in both nominal and uncertain platooning scenario.

Modeling eating disorders in young people

A mathematical model is proposed to simulate the eating disorders of bulimia or anorexia. Earlier models are extended to incorporate the body mass index, which plays a key role in the eating attitude of self thinners. The global existence and ultimate boundedness of solutions of the nonlocal model are proved by using estimates of solutions. The basic reproduction number of eating disorder contagion is shown to be the invasion threshold. The testable linear and nonlinear stability conditions are established by Lyapunov functions. Further numerical simulations are given to reveal how self-forces and peer pressures to be thinner affect the emergence and distributions of eating disorders.

An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

Does Unemployment Induce Crime in Society? A Mathematical Study

Today unemployment has become a global phenomenon which may be instrumental in forcing unemployed persons to earn their livelihood in an illegal manner resulting in a crime. It is possible that unemployed individuals may become more prone to develop a tendency of committing a crime when they come in contact with persons involved in criminal activities but are still unexposed. Further, when unemployed individuals are exposed to have committed a crime, they are captivated and finally awarded imprisonment if the offence charged on them is proved under the existing criminal laws. In this paper, a nonlinear mathematical model is developed to study the role of unemployment in inducing crime by taking into account four dependent variables representing the unemployment class, the employment class, the criminal class and the jail class. The model analysis, using stability theory of ordinary differential equations, provides some local and nonlinear stability conditions regarding stability of equilibrium of the model system. It is inferred that the endemic equilibrium is locally asymptotically stable as well as nonlinearly stable. Numerical simulations of the model system have also been carried to support the analytical findings and showing the effect of certain key parameters on different variables. It is observed that the increase in unemployment rate induces crime in the community leading to increase the burden on jail class.

Thermal convection in an inclined porous layer with Brinkman law

A model for thermal convection of a fluid saturating an inclined layer of porous medium with a Brinkman law and stress-free boundary conditions is studied. When the Darcy number $$\tilde{D}a$$ is zero, this problem has been studied by Rees and Bassom (Acta Mech 144(1–2):103–118, 2000). When the Brinkman term is present in the model ( $$\tilde{D}a\not =0$$ ) the basic motion is a combination of hyperbolic and polynomial functions. With the Chebyshev collocation method we study the linear instability of the basic motion for three-dimensional perturbations. We also give nonlinear stability conditions and, for longitudinal perturbations, we prove the coincidence of linear and nonlinear critical Rayleigh numbers.

On the dynamics of an intraguild predator–prey model

An intraguild predator–prey model with a carrying capacity proportional to the biotic resource, is generalized by introducing a Holling type II functional response. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. The existence of biologically meaningful equilibria (boundary and internal equilibria) has been investigated. Linear and nonlinear stability conditions for biologically meaningful equilibria are performed. Finally, numerical simulations on different regimes of coexistence and extinction of the involved populations have been shown.

Spectra and Stability of Spatially Periodic Pulse Patterns II: The Critical Spectral Curve

In the stability analysis of pulse patterns in singularly perturbed reaction-diffusion systems, the scale separation is often exploited to reduce complexity. There are various methods to decompose the spectrum of the linearization about the pattern into slow and fast pieces and thus gain control over the spectrum in the asymptotic limit. These reduction methods have been developed in prototype models such as the FitzHugh--Nagumo, Gray--Scott, and Gierer--Meinhardt equations, which are of slowly linear nature; i.e., the slow dynamics away from the pulses are driven by linear equations. Recently, these methods were extended to homoclinic pulses in a general, slowly nonlinear class of reaction-diffusion systems. Yet, a straightforward extension to periodic pulse solutions is impossible due to two major obstructions. First, the reduction methods that lead to asymptotic spectral control for the homoclinic pulses fail for periodic patterns in slowly nonlinear systems. Second, there is a curve of spectrum attached to the origin that shrinks to the origin in the asymptotic limit, whereas for homoclinic pulses only a simple eigenvalue resides at the origin. Therefore, controlling the spectrum in the asymptotic limit is insufficient to decide upon stability. The first obstruction has been addressed in the companion paper [B. de Rijk, A. Doelman, and J. D. M. Rademacher, SIAM J. Math. Anal., 48 (2016), pp. 61--121]. In this paper we focus on the second obstruction: we obtain leading-order control over the critical spectral curve attached to the origin. The proof relies on Lin's method and utilizes exponential trichotomies. Our results yield explicit conditions for nonlinear stability and instability for stationary, spatially symmetric, periodic pulses in a large class of multicomponent, slowly nonlinear, reaction-diffusion systems. Moreover, we gain insight into the associated destabilization mechanisms.