Stability Of Steady-state Solutions Research Articles

A Thin Film Model for Meniscus Evolution

In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of liquid, i.e. a liquid contained between two solid surfaces and part of the liquid surface is in contact with the air. The liquid is governed by Navier–Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the liquid thickness (lubrication approximation) and the contact angle between the liquid–solid and the liquid–gas interfaces close to π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}. This resulting model is a free boundary problem for the equation ht+(h3hxxx)x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_t + (h^3h_{xxx})_x = 0$$\\end{document}, for which we have h>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h>0$$\\end{document} at the contact point (different from the usual thin film equation with h=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h=0$$\\end{document} at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Furthermore, the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from the no-slip condition. Additionally, we show the global stability of steady state solutions in a periodic setting.

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

Including the parallel mass flow in calculating the steady-state solutions and stability of the momentum balance equations for a quasisymmetric stellarator

The Helically Symmetric Experiment (HSX) is a quasisymmetric stellarator with minimal parallel viscous damping in a helical direction. The parallel flow (Vǁ) along the magnetic field is similarly weakly damped by viscosity. In this paper, the self-consistent steady-state parallel and poloidal momentum balance equations are used to show that a large Vǁ on the order of the ion thermal velocity can increase the ion resonant radial electric field (Er) beyond the value calculated using the typical approximation that Vǁ is zero. By altering the damping of Vǁ, either by degrading the quasisymmetry or varying the neutral density, the ion resonant Er can shift in a controllable fashion. It is shown explicitly that there exist stable and unstable steady-state solutions in the two-dimensional space of Vǁ and Er. A stability analysis of each solution is performed by calculating the eigenvalues and eigenvectors of the Jacobian. The unstable solution corresponds to a saddle point in which the eigenvalues have opposite signs. The analysis leads to the conclusion that unstable solutions occur when the derivative of the total poloidal damping with respect to Er is positive. A hysteresis in Er and Vǁ is observed when the radial current density is linearly increased to a maximum and then decreased back to zero. Jumps in the radial electric field and the parallel flow are observed as the radial current density drives the evolution from one stable point to the next. This result is similar to experimental data observed on several devices.

Dissipative light bullets in a doped and weakly nonlocal optical fiber

The letter introduces an extended (3+1)-dimensional [(3+1)D] nonlocal cubic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in optical fiber amplifiers under the interplay between dopants and a spatially nonlocal nonlinear response. The model equation includes the effects of fiber dispersion, linear gain, nonlinear loss, fiber nonlinearity, atomic detuning, linear and nonlinear diffractive transverse effects, and nonlocal nonlinear response. A system of coupled ordinary differential equations for the amplitude, temporal, and spatial pulse widths and position of the pulse maximum, unequal wavefront curvatures, chirp parameters, and phase shift is derived using the variational technique. A stability criterion is established, where a domain of dissipative parameters for stable steady-state solutions is found. Direct integration of the proposed nonlocal evolution equation is performed, which allows us to investigate the evolution of the Gaussian beam along a doped nonlocal optical fiber, showing stable self-organized dissipative spatiotemporal light bullets.

Multiplicity of solutions of lifted jet edge flames: Symmetrical and non-symmetrical configurations

A numerical investigation of the combustion process taking place after fuel and oxidizer are injected separately into a planar channel from an end-wall porous plug is presented. A fuel stream injected in the middle of the channel is surrounded on both sides by oxidizer streams and two edge flames are formed after contact of reactants and ignition. The formulation of the problem is symmetrical with respect to the middle of the channel. The study is based on the coupled Navier-Stokes and transport equations with one-step Arrhenius-type combustion kinetics. The main focus is on the influence of the fuel Lewis number, the flow rate and the Damköhler number on the flame structure. It is shown that symmetrical and non-symmetrical configurations of edge flames are possible for the same set of parameters, and that these solutions can be simultaneously stable. At the same time, the areas of existence of symmetric and non-symmetric flame configurations turn out to be different. Further, it is demonstrated that there can be at least seven different (but not all stable) steady-state solutions for the same set of parameters, although the evidence for the absence of other solutions, in addition to those found, cannot be achieved within the framework of the methods used in the study. These results may be critical for flames with low fuel Lewis number (e.g. hydrogen flames) and highlight the importance of taking into account the possibility of non-symmetrical flame configurations in the design and operation of combustion devices based on diffusion flames.

Magnetoelastic simultaneous resonance of axially moving plate strip under a line load in stationary magnetic field

This paper studies the primary-internal simultaneous resonance of the moving ferromagnetic thin plate with the action of an alternating line load in a transverse stationary magnetic field. Based on the theory of thin plates, expressions of strain energy and kinetic energy of the thin plate are given. According to the theoretical magnetoelastic model for ferromagnets, the calculation formula for electromagnetic force is derived. The magnetoelastic coupling nonlinear vibration equation is established by using Hamilton’s principle. Duffing-type nonlinear differential equations are obtained for the clamped-hinged plate strip by applying the Galerkin method. The steady-state response of the system with primary resonance and 1:3 internal resonance is given by using the method of multiple scales, and the stability of steady-state solutions are analyzed according to the theory of Lyapunov stability. The primary-internal simultaneous resonance characteristics of the plate strip are presented through numerical examples. Results show that: energy exchange between the first two modes is available and the first mode dominates; jump phenomena exist in amplitude–frequency curves; both the values of steady solutions and the multi-valued regions of amplitude change with varieties of physical parameters; and numerical results agree with analytical results, which verifies the reliability of analytical research method in this paper.

A compound double pendulum with friction

We study a version of the two-degree-of-freedom double pendulum in which the two point masses are replaced by rigid bodies of irregular shape and nonconservative forces are permitted. We derive the equations of motion by analysing the forces involved in the framework of screw theory. This distinguishes the work from similar studies in the literature, which typically consider a double pendulum composed with rods and assume equations of motion without derivation. The equations of motion are solved numerically using the fourth-order Runge-Kutta method to show that decreasing the friction of the axles can cause the trajectory of one of the pendulums to become aperiodic. The stability of steady state solutions is also analysed.

On a hyperbolic-parabolic chemotaxis system.

Stability of steady state solutions associated with initial and boundary value problems of a coupled fluid-reaction-diffusion system in one space dimension is analyzed. It is shown that under Dirichlet-Dirichlet type boundary conditions, non-trivial steady state solutions exist and are locally stable when the system parameters satisfy certain constraints.

DYNAMICS OF A FISHERY MODEL WITH CONTINUOUS THRESHOLD HARVESTING POLICY AND ITS LEVERAGE FOR CONSERVATION AND MANAGEMENT

There is a global decline in marine fish abundance due to unsustainable harvesting. An effective harvesting policy can protect the overfished population from possible extinction. In this study, we used a mathematical model characterized by density-dependent refuge protection for herbivorous fish, exhibiting an anti-predator response in presence of a generalist invasive fish. The anti-predator behavior entails predator density-dependent reduced fecundity of the herbivorous fish. The model assumes a continuous threshold harvesting policy (CTHP) for the herbivorous fish and uses the catch-per-unit-effort (CPUE) hypothesis for harvesting the invasive fish. The CTHP allows harvesting of the herbivorous fish only when the density of the herbivorous fish exceeds a specified threshold value, thus ensuring the long-term sustainability of the herbivorous fish stock. The existence and stability of steady-state solutions and the bifurcations of the model are investigated. Our study reveals that the level of apprehension of the herbivorous fish and fishing efforts will play a significant role in the stability of the system. We examine the existence of the bionomic equilibrium and then study the dynamic optimization of the harvesting policy by employing Pontryagin’s maximum principle. We discuss different subsidies and tax policies for the effective management of a sustainable fishery. We use numerical simulations to compare the revenues corresponding to the harvest policies based on maximum sustainable yield (MSY), maximum economic yield (MEY), and optimal sustainable yield (OSY) for inferring an ecologically sustainable and economically viable harvesting policy.

Nonlinear analysis of the delayed tyre model with control-based continuation

In this study, the numerical bifurcation analysis of a shimmying wheel is performed with a non-smooth, time-delayed model of the tyre-ground contact. This model is capable of reproducing the bistable behaviour often observed in experiments: a stable equilibrium and a stable periodic orbit coexisting for the same set of system parameters, that the simpler quasi-steady tyre models fail to capture. In the bistable parameter domain, there also exists an unstable periodic orbit within the separatrix between the domains of attractions of the two stable steady-state solutions. Although this solution never appears in a real-life system, one may still gain valuable information from tracing it as it gives an indication about the level of perturbation that would drive the system from one stable solution to the other. However, the complexity of the laws governing partial sticking and sliding in the tyre-ground contact makes the numerical bifurcation analysis with the traditional, collocation-based techniques infeasible. Instead, this study is based on numerical simulations and the technique of control-based continuation (CBC) to track the stable and unstable periodic solutions of the system allowing for the assessment of the accuracy of the non-smooth, delayed tyre model in replicating the dynamics observed in experiments. In the meantime, the physics-based model provides an insight into the relationship between the sticking and sliding regions appearing in the tyre-ground contact and the global dynamics of the system.

Dynamics of a predator–prey system with nonlinear prey-taxis

In this paper, we investigate a predator–prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunov–Schmidt reduction method.

Intricate and multiple chirped waves geometric structures solutions of two-mode KdV equation, spectral and stability analysis

In a fluid, when two modes are propagating simultaneously in the same direction with different velocities ([Formula: see text] and [Formula: see text]), they are regarded as first and second modes. The two-mode Korteweg de Vries equation (TMKdVE) has been currently studied in numerous works in the literature. However, the distinction between the two modes was not studied. Here, exact solutions of the TMKdVE are obtained by using the unified method. It is found that each solution depends either on [Formula: see text] or on [Formula: see text]. These solutions are taken to assign to the first mode (FM) and the second mode (SM), respectively. These results are completely new. In this work, attention is focused on investigating the different chirped wave structures. It is shown that intricate, complex and multiple chirped waves in the two modes occur. Furthermore, it is shown that each mode can be seen in two visions: right wave (RW) and left wave (LW). The stability of steady-state solution is analyzed, where it is found that it is stable against varying the velocity [Formula: see text]. While when varying the velocity [Formula: see text], it is unstable when [Formula: see text] and stable when [Formula: see text]. The properties of the waves, the characteristic wavelength, frequency, traveling speed and spectrum content are shown graphically.

Competing feedback in an idealized tide-influenced delta network

The morphodynamic evolution of river deltas is intimately tied to flow and sediment partitioning at bifurcations. In this work, the long-term equilibrium configuration of a simple delta network is investigated by means of an analytical model, which accounts for the effect of small tidal oscillations. Differently from individual bifurcations where tidal action is always a stabilizing factor, in the case of a tree-like delta with multiple bifurcations a dual response emerges. Specifically, depending on the values of four reference parameters functions of tidal amplitude, upstream flow conditions, and channels geometry, tides can either promote or discourage an unbalanced discharge distribution. This behavior primarily concerns the apex bifurcation, which is affected by the variations of the relative tidal amplitude at the internal nodes. In turn these variations depend on how flow and sediment are diverted upstream. The stability of steady-state solutions is found to be governed by the sign and magnitude of the slope asymmetry between channels. This work provides a basic modeling framework for the interpretation of the autogenic response of multiple coupled micro-tidal bifurcations, which can potentially be extended to include in a unified scheme erosional and depositional processes typical of fluvio-deltaic systems.

Relaxation to Equilibrium in the One-Dimensional Thin-Film Equation with Partial Wetting and Linear Mobility

We investigate the large time behavior of compactly supported smooth solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. Relaxation rates are obtained for initial data which are close to a steady state in a suitable sense. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

Parameter analyses of suspended cables subjected to simultaneous combination, super and sub-harmonic excitations

The present study is dealing with a geometrically nonlinear model of the suspended cable subjected to multi-frequency excitations. In particular, two of the super, sub-harmonic, and combination resonances are excited simultaneously here. The nonlinear integro-differential equations of the suspended cable are introduced, together with the Galerkin method, to obtain a reduced-order model, whose responses are investigated by solving the reduced ordinary differential equations. Then, the obtained single-mode discretization equations are solved using the method of multiple scales in the frequency regions of three simultaneous resonant cases, with the stability characteristics determined. Effects of parameters on resonance characteristics are carried out by investigating several numerical examples. Numerical results demonstrate that the two-frequency excitation has significant influences on the dynamical behaviors of the nonlinear system. Each harmonic excitation component's contribution to the overall resonant responses is mainly dependent on its excitation amplitude. The stable steady-state solutions are confirmed by using numerical integration, and the number of steady-state solutions varies from one to seven as to different parameters of the system in simultaneous resonances. Besides, it is of great significance to include the effects of excitation phase differences on nonlinear vibration characteristics.

Stable Steady-State Solutions of Some Biological Aggregation Models

Aggregation phenomena occur across the biological sciences, from cell adhesion to insect swarms, animal home ranges to human cities. Understanding the mechanisms by which they may spontaneously emerge has therefore generated much interest from applied mathematicians. Partial differential equations (PDEs) with nonlocal advection offer a popular formalism for studying aggregations. However, the inherent nonlocality, often necessary for ensuring continuum models are well-posed, makes their study technically challenging. Here, we take a different approach by studying a discrete-space system that can be formally related to classical nonlocal PDE approaches via a limiting procedure. We show how to find expressions for the asymptotically stable steady-states of this discrete-space system via an energy functional approach. This allows us to predict the size of aggregations as a function of the underlying movement mechanisms of individual organisms. We apply this to a recent model of cell adhesion, revealing a hysteresis property whereby the existing aggregations may persist even as the adhesion tendency decreases past the bifurcation point. We compare this to numerical solutions of the associated nonlocal PDE system, showing that the hysteresis property predicted by the discrete-space expressions is also present in the continuum system.

Compressible Gas Flow past a Plate with Upstream-Moving Surface

The numerical solution of the problem of compressible gas flow past a plate with upstream-moving surface is obtained. Unstable time-periodic and stable steady-state solutions of this problem are discussed. A new type of unsteady periodic flow in which the varying characteristics turn out to be integrally asymmetric at transonic velocities is obtained.

On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions

It is a well-known fact that the onset of Rayleigh–Benard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Benard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Benard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $$y= \pm H$$ and horizontal boundaries located at $$z=0$$ and $$z=d$$ . Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $$\delta = H/d$$ takes a certain value. In the limit $$H \rightarrow \infty $$ , we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $$2 \pi /\delta $$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $$\delta $$ , is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model

AbstractOver 50 years since Lorenz’s 1963 study and a follow-up presentation in 1972, the statement “weather is chaotic” has been well accepted. Such a view turns our attention from regularity associated with Laplace’s view of determinism to irregularity associated with chaos. In contrast to single-type chaotic solutions, recent studies using a generalized Lorenz model (GLM) have focused on the coexistence of chaotic and regular solutions that appear within the same model using the same modeling configurations but different initial conditions. The results, with attractor coexistence, suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. In this study, based on the GLM, we illustrate the following two mechanisms that may enable or modulate two kinds of attractor coexistence and, thus, contribute to distinct predictability: 1) the aggregated negative feedback of small-scale convective processes that can produce stable nontrivial equilibrium points and, thus, enable the appearance of stable steady-state solutions and their coexistence with chaotic or nonlinear oscillatory solutions, referred to as the first and second kinds of attractor coexistence; and 2) the modulation of large-scale time-varying forcing (heating) that can determine (or modulate) the alternative appearance of two kinds of attractor coexistence. Based on our results, we then discuss new opportunities and challenges in predictability research with the aim of improving predictions at extended-range time scales, as well as subseasonal to seasonal time scales.