Nonlinear Stability Criterion Research Articles

Analyzing the Stability of a Connected Moving Cart on an Inclined Surface with a Damped Nonlinear Spring

This paper examines the stability behavior of the nonlinear dynamical motion of a vibrating cart with two degrees of freedom (DOFs). Lagrange’s equations are employed to establish the mechanical regulating system of the examined motion. The proposed approximate solutions (ASs) of this system are estimated through the use of the multiple-scales method (MSM). These solutions are considered novel as the MSM is being applied to a new dynamical model. Secular terms have been eliminated to meet the solvability criteria, and every instance of resonance that arises is categorized, where two of them are examined concurrently. Therefore, the modulation equations are developed based on the representations of the unknown complex function in polar form. The solutions for the steady state are calculated using the corresponding fixed points. The achieved solutions are displayed graphically to illustrate the impact of manipulating the system’s parameters and are compared to the numerical solutions (NSs) of the system’s original equations. This comparison shows a great deal of consistency with the numerical solution, which indicates the accuracy of the applied method. The nonlinear stability criteria of Routh–Hurwitz are employed to assess the stability and instability zones. The value of the proposed model is exhibited by its wide range of applications involving ship motion, swaying architecture, transportation infrastructure, and rotor dynamics.

Inspection of the nonlinear instability of electrified Casson fluids: a novel approach

The nonlinear electrohydrodynamic (EHD) instability of a circular cylindrical interface, separating two Casson prototypes is the aim of the present inspection. The motivation for the issue is owing to the increasing significance of several engineering and practicle physics. Because of the complicated mathematical processes, the contribution is performed through the viscous potential theory (VPT). The methodology resulted in a nonlinear characteristic equation of the interface displacement. This equation is transformed to simulate a hybrid Rayleigh-Helmholtz Duffing oscillator (HRDO) with complex coefficients. Guaranteeing the stability of fluid interfaces is crucial in industrial coating operations to provide consistent and even coatings. Gaining comprehension of the nonlinear stability criterion helps to enhance the management of coating thickness and uniformity. A novel methodology via the Non-perturbative approach (NPA) is employed. Through the case of the real coefficients, the later form is validated using a numerical solution (NS). It is concluded that the Ohnesorge contribution has a destabilizing effect on the stability zone in contrast with permeability factor. The electrical coefficients are found to be decaying factors in the stability configuration. It is also established that the viscosity decays the nonlinear steadiness requirement of the structure. The approach is a simple, promising, effective, and interesting.

An extended Hasegawa–Mima equation for nonlinear drift wave turbulence in general magnetic configurations

We propose an extended Hasegawa–Mima equation describing the evolution of nonlinear drift wave turbulence in general magnetic configurations. Such HMGM equation can be derived within the kinetic framework of guiding center motion or from a two-fluid model of an ion-electron plasma by application of a drift wave turbulence ordering that does not involve conditions on spatial derivatives of magnetic field and plasma density. The HMGM equation is therefore appropriate to describe the evolution of drift wave turbulence in strongly inhomogeneous magnetized plasmas, such as magnetospheric and stellarator plasmas, involving complex magnetic field geometries and non-uniform plasma density distributions. We find conservation laws (mass, energy, and generalized enstrophy) of the HMGM equation, study its algebraic (Hamiltonian) structure, and prove a nonlinear stability criterion for steady solutions through the energy-Casimir method. We then apply these results to describe drift waves and infer the existence of stable toroidal zonal flows with radial shear in dipole magnetic fields.

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.

Nonlinear stability of two dusty magnetic liquids surrounded via a cylindrical surface: impact of mass and heat spread

The current article examines a nonlinear axisymmetric streaming flow obeying the Rivlin–Ericksen viscoelastic model and overloaded by suspended dust particles. The fluids are separated by an infinite vertical cylindrical interface. A uniform axial magnetic field as well as mass and heat transmission (MHT) act everywhere the cylindrical flows. For the sake of simplicity, the viscous potential theory (VPT) is adopted to ease the analysis. The study finds its significance in wastewater treatment, petroleum transport as well as various practical engineering applications. The methodology of the nonlinear approach is conditional primarily on utilizing the linear fundamental equations of motion along with the appropriate nonlinear applicable boundary conditions (BCs). A dimensionless procedure reveals a group of physical dimensionless numerals. The linear stability requirements are estimated by means of the Routh–Hurwitz statement. The application of Taylor’s theory with the multiple time scales provides a Ginzburg–Landau equation, which regulates the nonlinear stability criterion. Therefore, the theoretical nonlinear stability standards are determined. A collection of graphs is drawn throughout the linear as well as the nonlinear approaches. In light of the Homotopy perturbation method (HPM), an estimated uniform solution to the surface displacement is anticipated. This solution is verified by means of a numerical approach. The influence of different natural factors on the stability configuration is addressed. When the density number of the suspended inner dust particles is less than the density number of the suspended outer dust particles, and vice versa, it is found that the structure is reflected to be stable. Furthermore, as the pure outer viscosity of the liquid increases, the stable range contracts, this means that this parameter has a destabilizing effect. Additionally, the magnetic field and the transfer of heat don’t affect the nature of viscoelasticity.

Nonlinear EHD instability of two viscoelastic fluids under the influence of mass and heat transfer

This study attempts to provide an approach to studying the nonlinear stability of a vertical cylindrical interface between two Oldroyd-B prototypes. An unchanged axial electric field influences the system, and porous medium, and the effects of heat and mass transfer (MHT) are considered. Hsieh's modulation and the viscous potential flow (VPT) are used to abbreviate the mathematical analysis. The viscoelastic Oldroyd-B model significant role in geothermal, engineering and industrial enhancement motivated us to carry out this in-depth investigation. The methodology of the nonlinear technique depends mainly on solving the linear equations of motion and applying the appropriate nonlinear boundary conditions. Numerous non-dimensional physical numbers are exposed using a non-dimensional technique. The stability conditions are theoretically achieved and numerically verified. As a limiting case, the linear dispersion equation is accomplished, and a set of stability diagrams is reachable. Together with the nonlinear stability method, a Ginzburg–Landau equation is derived. Subsequently, both theoretical and numerical methods are used to achieve the nonlinear stability criteria. Furthermore, a precise perturbed approach for surface deflection is achieved theoretically and numerically using the Homotopy perturbation method and the extended frequency conception. Along with the linear approach, it is found that the structure becomes unstable by the Laplace, Reynolds, Weber, and elasticity quantities as well as the linear MHT parameter. Furthermore, the stability zones are enhanced in the nonlinear instability approach.

ON THE NONLINEAR STABILITY FOR QUASI-GEOSTROPHIC FORCED ZONAL FLOWS

This paper continues a series of studies providing stability crite¬ria for quasigeostrophic forced zonal flows in in the presence of lateral diffusion and bottom dissipation of the vertical vorticity. We study the Lyapunov stability of a stationary and longitude independent ba¬sic flow, obtaining linear and nonlinear stability criteria expressed in terms of the maximum shear of the basic flow and/or its meridional derivative, extending some previous results.

Numerical prediction of transient electrohydrodynamic instabilities under an alternating current electric field and unipolar injection

In this paper, a direct numerical simulation (DNS) of dielectric fluid flow subjected to unipolar injection under an alternating current (AC) electric field is carried out. The effect of frequency f of pulsed direct current (PDC) and AC on the transient evolution of electroconvection and their subcritical bifurcations are investigated in details. Electroconvection under PDC or AC tends to exhibit oscillating flow due to the periodic boundary condition of charge density and potential compared to the direct current (DC) case. The results demonstrate that under the PDC field, the linear criterion Tc decreases with increasing frequency, while the nonlinear stability criterion Tf is hardly affected. Under the AC field, a critical frequency fc = 0.0316 is found, which separates electroconvection into two typical flow regimes—periodic flow regime (f < fc) and inhibited flow regime (f ≥ fc)—depending on whether free charges can reach the collector electrode before electric field inversion. AC-electrohydrodynamics (EHD) systems promote various flow patterns with relatively lower voltage regimes than DC-EHD systems. These mechanisms of electroconvection under the PDC/AC field offer unique possibilities for fluid flow control in biological EHD-driven flow and portable EHD applications.

Nonlinear EHD Instability of Two-Superposed Walters’ B Fluids Moving through Porous Media

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across the interface. Additionally, the structure is pervaded via a uniform, normal electrical field in the absence of superficial charges. The nonlinear scheme basically depends on analyzing the linear principal equation of motion, and then applying the appropriate nonlinear boundary-conditions. The current organization creates a nonlinear characteristic equation describing the amplitude performance of the surface waves. The classical Routh–Hrutwitz theory is employed to judge the linear stability criteria. Once more, the implication of the multiple time scale with the aid of Taylor theory yields a Ginzburg–Landau equation, which controls the nonlinear stability criteria. Furthermore, the Poincaré–Lindstedt technique is implemented to achieve an analytic estimated bounded solution for the surface deflection. Many special cases draw upon appropriate data selections. Finally, all theoretical findings are numerically confirmed in such a way that ensures the effectiveness of various physical parameters.

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

Models of double-diffusive convection in a fluid layer with the viscosity having linear dependence on the temperature are analyzed here. The density is assumed to have a quadratic dependence on temperature and a linear dependence on the species concentration. Conditional and unconditional nonlinear energy stability criteria for two different models are derived. A comparison of linear and nonlinear theories suggests that there are significant regions where the basic state is subject to a deep subcritical instability when the value of solute Rayleigh number is high.

A Nonlinear Azimuthal Instability of Hydromgantic Rigid-Rotating Column

This work examines a new approach to studying the nonlinear azimuthal instability analysis. The system consists of two rotating fluids through porous media in the influence of a uniform azimuthal magnetic field. For gullibility, the problem is assumed in a planar configuration. The boundary-value problem reveals a differential equation of nonlinearity nature which controls the surface deflection of the interface. The investigation of this equation is based mainly on the homotopy perturbation technique. The linear and nonlinear stability criteria are conducted. Besides, the profile of the surface deflection is theoretically achievable. The numerical calculations are done to display the effect of the several physical parameters on the stability profile. It is found that the ratio of the densities between the two fluid columns plays an interesting role in the stability picture in linear as well as the nonlinear approaches. For instance; a dual role of the density ratio occurs when the density of the inner column is greater than that of the outer one. Furthermore, the azimuthal wavenumber, like the axial wavenumber, plays a stabilizing influence.

Dual Compound–Compound Synchronization of Twelve n-Dimensional Dynamical Systems

A dual synchronization scheme combined with compound–compound synchronization is proposed. In usual compound–compound synchronization scheme, synchronization has been achieved for six dynamical systems, while in the presented scheme compound–compound synchronization has been developed in the dual synchronization frame involving twelve dynamical systems. Numerous synchronization schemes such as dual compound, dual combination, compound–compound and compound synchronization may be derived as particular cases of the presented scheme. By employing nonlinear control method and Lyapunov stability criteria, controllers are designed. Illustrations are presented using Genesio-Tesi and Wang chaotic systems. Computational results are presented to show the effectiveness of the proposed methodology. A good agreement has been obtained between analytical methodology and numerical treatment.

Available energy of magnetically confined plasmas

The concept of the available energy of a collisionless plasma is discussed in the context of magnetic confinement. The available energy quantifies how much of the plasma energy can be converted into fluctuations (including nonlinear ones) and is thus a measure of plasma stability, which can be used to derive linear and nonlinear stability criteria without solving an eigenvalue problem. In a magnetically confined plasma, the available energy is determined by the density and temperature profiles as well as the magnetic geometry. It also depends on what constraints limit the possible forms of plasma motion, such as the conservation of adiabatic invariants and the requirement that the transport be ambipolar. A general method based on Lagrange multipliers is devised to incorporate such constraints in the calculation of the available energy, and several particular cases are discussed for which it can be calculated explicitly. In particular, it is shown that it is impossible to confine a plasma in a Maxwellian ground state relative to perturbations with frequencies exceeding the ion bounce frequency.

Solitary waves of two-layer quasi-geostrophic flow and analytical solutions with scalar nonlinearity

In this paper we investigate the instability of two-layer quasi-geostrophic model with scalar nonlinearity, which is a geophysical fluid dynamics model, and give a nonlinear stability criterion. Furthermore, the analytical solutions of the two-layer quasi-geostrophic model are obtained via the Jacobi elliptic periodic function transform method and symbolic computation. It is efficient in solving a range classes of nonlinear partial differential equations for this method. The result shows that the structure of Rossby waves is influenced by many factors such as base flow and scalar nonlinearity.

Multi-switching compound-compound anti-synchronization of non-identical hyperchaotic systems

ABSTRACT In this paper, a synchronization scheme for six non-identical hyperchaotic systems is investigated. The paper addresses a synchronization scheme of five leader (master) and one follower (slave) non-identical hyperchaotic systems. This synchronization scheme is first of its kind and different from existing schemes as instead of a scaling system a compound scaling signal is used. The presented scheme is termed as ‘multi-switching compound-compound anti-synchronization’ (MCCAS). Asymptotically stable synchronized states are established between leader and follower systems by employing nonlinear control method and Lyapunov stability criteria. Additional leader systems and hyperchaotic systems enhance the complexity of this synchronization scheme which is useful in secure communication. The information signal can be split and loaded with different signals of leader systems to increase the grade of security. To justify the theoretical results, an example is considered where Pang-Liu, Zheng, Chen, Liu and Newton-Leipnik hyperchaotic systems are considered as leader systems and Lorenz hyperchaotic system is considered as a follower system. Theoretical results are confirmed by graphical analysis.

Instability of catenary-type flexible risers conveying fluid in subsea environments

As water depths for oil and gas exploration and extraction increase, structures such as flexible risers, mooring lines, and umbilical cables are increasingly being used for subsea environments. Compared to conventional fixed-type structures and vertical risers, the dynamics of flexible risers is significantly more complex. In particular, the flexible structures may be prone to dynamic instabilities. The goal of the present paper is to provide a comprehensive study of the dynamics, stability, and vibration of flexible risers. We use Kirchhoff's theory of an extensible, flexible rod that resists torsion to develop a set of nonlinear equations for the dynamics of risers. The resulting model incorporates drag and the effects of the fluid being transported internally. Using a nonlinear stability criterion, our analyses show the nonlinear stability of a simple catenary-type riser modeled either as an inextensible or extensible string. For the more advanced rod models, we use a linear stability analysis to show how the internal fluid being conveyed can destabilize certain static configurations.

Sharp Nonlinear Stability Criterion of Viscous Non-resistive MHD Internal Waves in 3D

We consider the dynamics of two layers of incompressible electrically conducting fluid interacting with a magnetic field, which are confined within a 3D horizontally infinite slab and separated by a free internal interface. We assume that the upper fluid is heavier than the lower fluid so that the fluids are susceptible to Rayleigh–Taylor instability, yet we show that the viscous and non-resistive problem around the equilibrium is nonlinearly stable provided that the strength of the vertical component of the steady magnetic field, $${|{\bar{B}_3}|}$$ , is greater than the critical value, $${\mathcal{M}_c}$$ , which we identify explicitly. We also prove that the problem is nonlinearly unstable if $${|{\bar{B}_3}| < \mathcal{M}_c}$$ . Our results indicate that the non-horizontal magnetic field has a strong stabilizing effect on the Rayleigh–Taylor instability but the horizontal one does not have this in 3D.

Multi-switching Synchronization of Four Non-identical Hyperchaotic Systems

The paper addresses a combination synchronization scheme achieved for different switches of three master and one slave hyperchaotic systems. An asymptotically stable synchronized state is derived for different switches of master systems and slave system by using nonlinear control method and Lyapunov stability criteria. To elaborate the presented scheme Pang–Liu hyperchaotic system, Zheng hyperchaotic system and Chen hyperchaotic system are considered as master systems and Newton–Leipnik hyperchaotic system is considered as slave system. Theoretical and graphical results converge to the same conclusion which proves the efficiency of the applied approach.

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $\Lambda$ of the basic flow crosses a critical threshold $\Lambda_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations.

Decentralised output-feedback control for a class of large-scale stochastic high-order upper-triangular nonlinear systems

ABSTRACTUnder the weaker conditions on the drift and diffusion terms, this paper focuses on the global decentralised output-feedback control for a class of large-scale stochastic high-order upper-triangular nonlinear systems. By introducing an appropriate coordinate transformation, the original system is transformed into an equivalent one with tunable gain. After that, by reasonably combing the homogeneous domination approach with stochastic nonlinear systems stability criterion, and skillfully choosing the low gain scale, the decentralised output-feedback controller is constructed for each subsystem to ensure that the closed-loop system is globally asymptotically stable in probability. The simulation example is given to demonstrate the effectiveness of the proposed design scheme.