Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier–Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic “galloping”, or “pulsating”, solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier–Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.     

Nonlinear stability of cylindrical shells subjected to axial flow: Theory and experiments

This paper, is concerned with the nonlinear dynamics and stability of thin circular cylindrical shells clamped at both ends and subjected to axial fluid flow. In particular, it describes the development of a nonlinear theoretical model and presents theoretical results displaying the nonlinear behaviour of the clamped shell subjected to flowing fluid. The theoretical model employs the Donnell nonlinear shallow shell equations to describe the geometrically nonlinear structure. The clamped beam eigenfunctions are used to describe the axial variations of the shell deformation, automatically satisfying the boundary conditions and the circumferential continuity condition exactly. The fluid is assumed to be incompressible and inviscid, and the fluid–structure interaction is described by linear potential flow theory. The partial differential equation of motion is discretized using the Galerkin method and the final set of ordinary differential equations are integrated numerically using a pseudo-arclength continuation and collocation techniques and the Gear backward differentiation formula. A theoretical model for shells with simply supported ends is presented as well. Experiments are also described for (i) elastomer shells subjected to annular (external) air-flow and (ii) aluminium and plastic shells with internal water flow. The experimental results along with the theoretical ones indicate loss of stability by divergence with a subcritical nonlinear behaviour. Finally, theory and experiments are compared, showing good qualitative and reasonable quantitative agreement.     

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.     

A nonanalytic,quasistationary approximation scheme applied to a model problem

A model problem designed to incorporate several of the features of slow-motion gravity is analyzed. In common with a previous model calculation the present work supports the idea that divergent integrals which arise in analytic quasistationary perturbation schemes are artifacts of these schemes and can be eliminated by using singular perturbation techniques, in which they get replaced by finite nonanalytic terms. The new features of the present model are (1) it illustrates the necessity of straining the outgoing null coordinate to eliminate logarithmic nonuniformities in nonlinear order for large radius, and (2) it is exactly solvable for the pure-frequency case, which gives an important check on the approximation scheme. Straining the null coordinate immediately introduces time-odd terms in the near zone which are larger than the standard linear-theory damping result. However, these terms are shown to be spurious in that they are canceled by higher-order contributions in the approximation scheme. Thus the standard linear-theory damping result holds in the model, even when nonlinear effects are considered.Supported in part by The National Science Foundation under grant No. PH 79-11664.     

Multiple car-following model of traffic flow and numerical simulation

On the basis of the full velocity difference (FVD) model, an improved multiple car-following (MCF) model is proposed by taking into account multiple information inputs from preceding vehicles. The linear stability condition of the model is obtained by using the linear stability theory. Through nonlinear analysis, a modified Korteweg-de Vries equation is constructed and solved. The traffic jam can thus be described by the kink--antikink soliton solution for the mKdV equation. The improvement of this new model over the previous ones lies in the fact that it not only theoretically retains many strong points of the previous ones, but also performs more realistically than others in the dynamical evolution of congestion. Furthermore, numerical simulation of traffic dynamics shows that the proposed model can avoid the disadvantage of negative velocity that occurs at small sensitivity coefficients λ in the FVD model by adjusting the information on the multiple leading vehicles. No collision occurs and no unrealistic deceleration appears in the improved model.     

Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables

A study is carried out on nonlinear multimodal galloping of suspended cables. A consistent model of a curved cable-beam, geometrically nonlinear and able to torque, recently formulated by the authors, is used. The model accounts for quasi-steady aerodynamic forces, including the effect of static swing of the cable and dynamic twist of the cross-section. Complementary solution methods are employed, namely, finite-difference and Galerkin spatial discretization, followed by numerical time-integration, or Galerkin spatial discretization in conjunction with Multiple Scale perturbation analysis. The different techniques are applied to a cable close to the first cross-over point, at which a number of internal resonances exist. Branches of periodic solutions and their stability are evaluated as functions of wind velocity. The existence of branches of quasi-periodic solutions, originating from narrow unstable intervals and propagating elsewhere, is also proved. Qualitative and quantitative results furnished by the different investigation tools are compared among them, and the importance of the various components of motion, accounted or neglected in the reduced models, is discussed.     

Stability of the (Two-Loop) Renormalization Group Flow for Nonlinear Sigma Models

We prove the stability of the torus, and with suitable rescaling, hyperbolic space under the (two-loop) renormalization group flow for the nonlinear sigma model. To prove stability we use similar techniques to Guenther et al. (Commun. Anal. Geom. 10:741–777, 2002), where the stability of the torus under Ricci flow was first established. The main technical tool is maximal regularity theory.     

Some aspects of the harmonic balance method applied to the clarinet

The clarinet has been extensively studied by various theoretical and experimental techniques. In this paper, the harmonic balance method (HBM), a numerical method mainly working in the frequency domain, has been applied to solve a simple nonlinear clarinet model consisting of a linear exciter (for the reed) nonlinearly coupled to a linear resonator with visco-thermal losses (for the pipe). A recent and improved implementation of the HBM for self-sustained instruments has allowed us to study the model theoretically when including dispersion in the pipe or mass and damping terms in the reed model. The resulting periodic solutions for the internal pressure spectrum and the corresponding playing frequency are shown to align well with previous theoretical and experimental knowledge of the clarinet. Finally, we present and briefly discuss a few (probably unstable) oscillation regimes both with the HBM and with a real clarinet.     

Stability analysis of nonlinear Roesser-type two-dimensional systems via a homogenous polynomial technique

<正>This paper is concerned with the problem of stability analysis of nonlinear Roesser-type two-dimensional(2D) systems.Firstly,the fuzzy modeling method for the usual one-dimensional(1D) systems is extended to the 2D case so that the underlying nonlinear 2D system can be represented by the 2D Takagi-Sugeno(TS) fuzzy model,which is convenient for implementing the stability analysis.Secondly,a new kind of fuzzy Lyapunov function,which is a homogeneous polynomially parameter dependent on fuzzy membership functions,is developed to conceive less conservative stability conditions for the TS Roesser-type 2D system.In the process of stability analysis,the obtained stability conditions approach exactness in the sense of convergence by applying some novel relaxed techniques.Moreover,the obtained result is formulated in the form of linear matrix inequalities,which can be easily solved via standard numerical software.Finally,a numerical example is also given to demonstrate the effectiveness of the proposed approach.     

A new fully nonisothermal coupled model for the simulation of ice sheet flow

A shallow ice thermocoupled model for the complex nonlinear polythermal ice sheet dynamics is proposed and solved by means of efficient numerical methods. A novelty is the obstacle problem formulation associated to a nonlinear integro-differential equation (with nonlocal temperature dependent coefficients) for the ice sheet profile. This formulation is motivated by the free boundary feature and the influence of the temperature on the profile (fully nonisothermal model). Concerning the temperature equation, a dynamically prescribed surface temperature, obtained from an Energy Balance model corrected by the altitude effect, is posed. As the profile and temperature equations are fully coupled, a nonlinear PDE system governing the upper ice sheet profile, the velocity field, the temperature and the basal stress is stated. In addition to the numerical difficulties associated to the new profile equation, several techniques have been considered for the numerical solution of the temperature, velocity and basal magnitudes. Discussions concerning the nonlinear dynamics of the different involved magnitudes and the improvement in their computed values with respect to previous works are also presented.     

Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: A fully nonlinear approach

Within the framework of the fully nonlinear water waves equations, we consider a Stokes wavetrain modulated by the Benjamin-Feir instability in the presence of both viscous dissipation and forcing due to wind. The wind model corresponds to the Miles’ theory. By introducing wind effect on the waves, the present paper extends the previous works of [6] and [7] who neglected wind input. It is also a continuation of the study developed by [9] who considered a similar problem within the framework of the NLS equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by [9] from a linear stability analysis. Furthermore, it is found that wind input goes in the subharmonic mode of the modulation whereas dissipation damps the fundamental mode of the initial Stokes wavetrain.     

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.     

A New Variational Approach to the Stability of Gravitational Systems

We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by several authors based on the pioneering work by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type.     

A New Car Following Model： Comprehensive Optimal Velocity Model

In this paper, we present a new car-following model, i.e. comprehensive optimal velocity model (COVM), whose optimal velocity function not only depends on the following distance of the preceding vehicle, but also depends on the velocity difference with preceding vehicle. Simulation results show that COVM is an improvement over the previous ones theoretically. Then, the stability condition of the model is obtained by the linear stability analysis, which has shown that the model could obtain a bigger stable region thanprevious models in the phase diagram. Through the nonlinear analysis, the Burgers, Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are derived for the triangular shock wave, the soliton wave, and the kink-antikink soliton wave. At the same time, numerical simulations are also carried out to show that the model could simulate these density waves.     

Soliton contribution to correlations in a system of coupled chains

A model representing a two- or a three-dimensional array of classical harmonic chains withnonlinear coupling between them is investigated. Physically real systems to which this model applies are discussed. The model exhibits soliton-like nonlinear modes. The influence of these nonlinear modes on the static and the dynamic correlation functions is calculated by generalizing techniques developed for strictly one-dimensional systems. In the static correlation functions these modes lead to minor quantitative changes only. In certain dynamic correlation functions, however, a central peak is found to occur due to the nonlinear modes. The total weight and the width of this peak are calculated for a real spin system.     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

Power-Law Inflation in Spacetimes Without Symmetry

We consider models of accelerated cosmological expansion described by the Einstein equations coupled to a nonlinear scalar field with a suitable exponential potential. We show that homogeneous and isotropic solutions are stable under small nonlinear perturbations without any symmetry assumptions. Our proof is based on results on the nonlinear stability of de Sitter spacetime and Kaluza-Klein reduction techniques.     

Nonlinear Measures to Evaluate Upright Postural Stability: A Systematic Review

Conventional biomechanical analyses of human movement have been generally derived from linear mathematics. While these methods can be useful in many situations, they fail to describe the behavior of the human body systems that are predominately nonlinear. For this reason, nonlinear analyses have become more prevalent in recent literature. These analytical techniques are typically investigated using concepts related to variability, stability, complexity, and adaptability. This review aims to investigate the application of nonlinear metrics to assess postural stability. A systematic review was conducted of papers published from 2009 to 2019. Databases searched were PubMed, Google Scholar, Science-Direct and EBSCO. The main inclusion consisted of: Sample entropy, fractal dimension, Lyapunov exponent used as nonlinear measures, and assessment of the variability of the center of pressure during standing using force plate. Following screening, 43 articles out of the initial 1100 were reviewed including 33 articles on sample entropy, 10 articles on fractal dimension, and 4 papers on the Lyapunov exponent. This systematic study shows the reductions in postural regularity related to aging and the disease or injures in the adaptive capabilities of the movement system and how the predictability changes with different task constraints.     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.