On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $\mathbf{M}\mathbf{q}'' + \mathbf{D}\mathbf{q}' + (\mathbf{K} + \bar{\mu}\mathbf{N})\mathbf{q} \in-\mathbf{R}\operatorname{Sign}(\mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $\bar{\mu}= \bar{\mu}_{\mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $\bar{\mu}$ . Depending on $\bar{\mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $\bar{\mu}_{\mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(\bar{\mu}- \bar{\mu}_{\mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.     

Evolution of Domains of Attraction of a Forced Beam with Two-Mode Interaction

When a dissipative dynamical system has multiple attractors, it is a task to determine and recognize the global domain of attraction of each attractor. In this paper we study the global behavior of a forced hinged-clamped beam with two-mode interaction. The equation of motion of the beam is reduced to four first-order autonomous ordinary differential equations. The system has an internal resonance condition of 2 31, where 1 and 2 denote natural frequencies of the first and second modes, respectively. When the excitation frequency is near 1, the system can have three equilibrium solutions, among which two are asymptotically stable and one is unstable. We examine how the domains of attraction of two stable equilibrium solutions evolve as the forcing frequency is varied across jump points. By using a special plane which contains all equilibrium points, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution helps us better understand the jump phenomenon.     

Local stabilization for unstable bilinear systems with input saturation

This paper is concerned with local stabilization for unstable bilinear systems with input saturation. Given a prespecified polytope $\mathcal{P}$ of the state space containing the zero equilibrium point, a linear state feedback is designed to guarantee that the closed-loop system is asymptotically stable in $\mathcal{P}$ and $\mathcal{P}$ is enclosed in the domain of attraction of the zero equilibrium point. Such sufficient conditions are derived via linear matrix inequalities (LMIs). Finally, an example illustrates the effectiveness of the proposed method.     

A sum of squares approach to robust PI controller synthesis for a class of polynomial multi-input multi-output nonlinear systems

This paper presents a new algorithmic method to design PI controller for a general class of nonlinear polynomial systems. Design procedure can take place on certain or uncertain nonlinear model of plant and is based on sum of squares optimization.The so-called density function is employed to formulate the design problem as a convex optimization program in the sum of squares form. Robustness of design is guaranteed by taking parametric uncertainty into account with an approach similar to that of generalized ${\mathcal {S}}$ -Procedure. Validity and applicability of the proposed methods are verified via numerical simulations. The method presented here for PI controller design is not based on local linearization and works globally. Derived stability conditions overcome several drawbacks seen in previous results, such as depending on a linearized model or a stable model. Furthermore, employing sum of squares technique makes it possible to derive stability conditions with least conservatism and directly design controller for polynomial affine nonlinear systems.     

Multistage γ-level \mathcal{H}_{\infty} control for input-saturated systems with disturbances

For input-saturated systems with disturbances, states in the domain of attraction cannot converge to the origin, but only to neighborhood around it. In order to design the smallest possible target invariant set and the largest possible domain of attraction, in this paper, we introduce a multistage γ-level $\mathcal{H}_{\infty}$ control for achieving a smaller target invariant set within a given $\mathcal{H}_{\infty}$ performance level and a larger domain of attraction than results obtained in previous studies. In particular, for the case in which the disturbances satisfy a matched condition, this paper introduces an $\mathcal{H}_{\infty}$ control with an extra control part to perfectly reject these disturbances despite the uncertainties; the introduction of the $\mathcal{H}_{\infty}$ control with an extra control part causes the target invariant set to shrink to the origin and the $\mathcal{H}_{\infty}$ performance level to become zero.     

Spanwise effects on instabilities of compressible flow over a long rectangular cavity

The stability properties of two-dimensional (2D) and three-dimensional (3D) compressible flows over a rectangular cavity with length-to-depth ratio of \(L/D=6\) are analyzed at a free-stream Mach number of \(M_\infty =0.6\) and depth-based Reynolds number of \(Re_D=502\). In this study, we closely examine the influence of three-dimensionality on the wake mode that has been reported to exhibit high-amplitude fluctuations from the formation and ejection of large-scale spanwise vortices. Direct numerical simulation (DNS) and bi-global stability analysis are utilized to study the stability characteristics of the wake mode. Using the bi-global stability analysis with the time-averaged flow as the base state, we capture the global stability properties of the wake mode at a spanwise wavenumber of \(\beta =0\). To uncover spanwise effects on the 2D wake mode, 3D DNS are performed with cavity width-to-depth ratio of \(W/D=1\) and 2. We find that the 2D wake mode is not present in the 3D cavity flow with \(W/D=2\), in which spanwise structures are observed near the rear region of the cavity. These 3D instabilities are further investigated via bi-global stability analysis for spanwise wavelengths of \(\lambda /D=0.5{-}2.0\) to reveal the eigenspectra of the 3D eigenmodes. Based on the findings of 2D and 3D global stability analysis, we conclude that the absence of the wake mode in 3D rectangular cavity flows is due to the release of kinetic energy from the spanwise vortices to the streamwise vortical structures that develops from the spanwise instabilities.     

Global Dynamics of a Time-Delayed Microorganism Flocculation Model with Saturated Functional Responses

In this paper, a time-delayed model of microorganism flocculation with saturated functional responses is presented. We first analyse the local dynamics of this model with bifurcations in parameter fields, and then prove the collection of microorganisms is sustainable as well as obtain an explicit eventual lower bound of microorganism concentration when threshold parameter \(R_{0}>1\). This model has a backward bifurcation if \(w<R_{0}<1\) under an additional condition, which implies that the microorganism-free equilibrium coexists with a microorganism equilibrium. In these cases, we establish some sufficient conditions for the global stability by using a variant of the Lyapunov–LaSalle theorem.     

Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

Global analysis of some epidemic models with general contact rate and constant immigration

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found. For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained. For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable ; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.     

Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

We address the global solvability of the Kuramoto–Sivashinsky equation in a rectangular domain \([0,L_1]\times [0,L_2]\) . We give sufficient conditions on the width \(L_2\) of the domain, depending on the length \(L_1\) , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.     

Global Existence and Exponential Stability of Spherically Symmetric Solutions to a Compressible Combustion Radiative and Reactive Gas

In this paper, we establish the global existence and exponential stability of spherically symmetric solutions in \({H^i\times H^i\times H^i\times H^i (i=\rm 1,2,4)}\) for a multi-dimensional compressible viscous radiative and reactive gas. Our global existence results improve those known results. Moreover, we establish the asymptotic behavior and exponential stability of global solutions on \({H^i\times H^i\times H^i\times H^i (i=\rm 1,2,4)}\). This result is obtained for this problem in the literature for the first time.     

Fine Level Set Structure of Flat Isometric Immersions

A result by Pogorelov asserts that C ¹ isometric immersions u of a bounded domain \({S \subset \mathbb R^2}\) into \({\mathbb {R}^3}\) whose normal takes values in a set of zero area enjoy the following regularity property: the gradient \({f := \nabla u}\) is ‘developable’ in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.     

Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We consider only the circular case (e = 0) but under the action of a broader family of singular potentials: α-homogeneous potentials, for \(\alpha \in (0, 2)\), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter α and the mass parameter β, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y. Long, X. Hu and S. Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.     

A Spectral Criterion for Stability of a Steady Viscous Incompressible Flow Past an Obstacle

We show that the question of stability of a steady incompressible Navier-Stokes flow \({\mathrm{V}}\) in a 3D exterior domain \({\Omega}\) is essentially a finite-dimensional problem (Theorem 3.2). Although the associated linearized operator has an essential spectrum touching the imaginary axis, we show that certain assumptions on the eigenvalues of this operator guarantee the stability of flow \({\mathrm{V}}\) (Theorem 4.1). No assumption on the smallness of the steady flow \({\mathrm{V}}\) is required.     

On Global Attraction to Stationary States for Wave Equations with Concentrated Nonlinearities

The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as \(t\rightarrow \pm \infty \) to stationary states. The attraction is caused by nonlinear energy radiation.     

The Vlasov–Poisson–Landau System in {\mathbb{R}^{3}_{x}}

For the Landau–Poisson system with Coulomb interaction in ${\mathbb{R}^{3}_{x}}$ R x 3 , we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.     

Global Asymptotic Stability for Oscillators with Superlinear Damping

A necessary and sufficient condition is established for the equilibrium of the damped superlinear oscillator $$x^{\prime\prime} + a(t)\phi_q(x^{\prime}) + \omega^2x = 0$$ to be globally asymptotically stable. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation $$u^{\prime} + \omega^{q-2}a(t)\phi_q(u) + 1 = 0$$ is divergent or convergent. Since this nonlinear differential equation cannot be solved in general, it can be said that the presented result is expressed by an implicit condition. Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, it is proved that a certain growth condition of a(t) guarantees the global asymptotic stability for the equilibrium of the damped superlinear oscillator.     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.     

Basic Reproduction Ratios for Periodic Compartmental Models with Time Delay

In this paper, we establish the theory of basic reproduction ratio \(R_0\) for a large class of time-delayed compartmental population models in a periodic environment. It is proved that \(R_0\) serves as a threshold value for the stability of the zero solution of the associated periodic linear systems. As an illustrative example, we also apply the developed theory to a periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of \(R_0\).