Landau Damping in the Kuramoto Model

We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class \({C^n}\) (\({n\geq 4}\)). A criterion for linear stability of the uniform stationary state is established which, for basic examples in the literature, is equivalent to the standard condition on the coupling strength. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate n) for every trajectory issued from a sufficiently small \({C^n}\) perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.     

On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.     

Formation of phase-locked states in a population of locally interacting Kuramoto oscillators

In this paper, we present an asymptotic formation of phase-locked states from the ensemble of Kuramoto oscillators with a symmetric and connected interaction topology. For a limited interaction topology that does not have an all-to-all interaction, Lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness. Thus, we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates. Our synchronization estimation method consists of two parts. First we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase-locked states using ?ojasiewicz gradient inequality. Second, we show that for the initial configurations lying in the half circle, the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear Gronwall?s differential inequality for the total phase variance.     

On Numerical Landau Damping for Splitting Methods Applied to the Vlasov–HMF Model

We consider time discretizations of the Vlasov–HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size.     

Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow

We study the asymptotic complete entrainment of Kuramoto oscillators with inertia on symmetric and connected network. We provide several sufficient conditions for the asymptotic complete entrainment in terms of initial phase-frequency configurations, strengths of inertia and coupling, and natural frequency distributions. For this purpose, we reinterpret the Kuramoto oscillators with inertia as a second-order gradient-like flow, and adopt analytical methods based on several Lyapunov functions to apply the convergence estimate studied by Haraux and Jendoubi [21]. Our approach does not require any spectral information of the graph associated with the given network structure.     

Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection

In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results.     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in‐plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.     

Uniform asymptotic stability for damped linear oscillators with variable parameters

In this paper, sufficient conditions for uniform asymptotic stability of the damped linear oscillators with variable coefficients are presented. The result of the present study can be applied to even the case of negative damping. The conditions presented herein are shown to be essential for uniform asymptotic stability.     

A geometric analysis of the Lagerstrom model problem

Lagerstrom's model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low Reynolds number by the method of matched asymptotic expansions. In this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems, in particular invariant manifold theory. As an essential part of the dynamics takes place near a line of non-hyperbolic equilibria, a blow-up transformation is introduced to resolve these singularities. This approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem. In particular, the source of the logarithmic switchback phenomenon is identified.     

离散时滞奇异摄动控制系统的稳定性分析

对含不确定性结构的奇异摄动时滞离散控制系统进行稳定性研究.通过设计一种新的Lyapunov Krasovskii泛函, 基于Lyapunov稳定性理论,在时滞依赖情形下, 采取交叉项界定技术、 线性矩阵分析方法并运用引理, 推出在零到奇异摄动上界的整个区间范围内系统渐近稳定,给出充分性的稳定性判据.之后,再对其进行理论加深和推广, 得到更加简洁性的推论, 可以借助于MATLAB工具箱进行求解.最后,用算例证明本文所得方法的优越性和可行性.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equations in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time-dependent, Hamiltonian, linearized dynamics around a carefully chosen one-parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.     

Chimera state and route to explosive synchronization

Transition to explosive synchronization is exhibited in networks of Kuramoto oscillators with a positive correlation between the oscillator dynamics and their inner topological structure encoded in the vertex degrees relations, shedding a light over the explosive critical phenomena. Here we study emergence of chimera states for the large amplitude oscillations when degree-frequency correlation is established only for the vertices with the highest degrees. For the strong coupling regime no simultaneous coexistence of coherence and incoherence signatures is observed. The connection between the network dynamics and the range and strength of coupling is elucidated through extensive analytical investigation, presenting realistic simulations of a scale-free neural network of Caenorhabditis elegans.     

From singularity theory to finiteness of Walrasian equilibria

The main result of this paper states that there exists a residual subset of the set of critical economies whose associated equilibria are finite in number. We also show that this subset does not contain any open set and therefore the result is the best possible for our choice of topology (compact-open topology). The proof rests on results and concepts from singularity theory.     

Stability for systems of 1-D coupled nonlinear oscillators

The stability of the null solution of different systems of differential equations describing the motion of 1-D coupled nonlinear oscillators is discussed. Under certain assumptions we derive some stability results. Specifically, in the case of coupled damped oscillators we obtain asymptotic stability of the null solution (see Theorem 3.1, Example 3.1, and Fig. 2), while in the case of partial lack of damping we only obtain convergence to zero of the solution components corresponding to damped oscillators (see Theorem 4.1, Example 4.1, and Fig. 5). In all cases, including the case of coupled undamped oscillators, we obtain uniform stability of the null solution.     

Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu–Hethcote–Levin conjecture. Furthermore, an affirmative answer to Li–Graef–Wang–Karsai’s problem for SEIR model with permanent immunity and varying total population size is given.     

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

This paper is motivated by the stability problem of nonconstant periodic solutions of time‐periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way.     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.