Exact Boundary Controllability on a Tree-Like Network of Nonlinear Planar Timoshenko Beams

This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions and,using the methods of quasilinear hyperbolic systems,prove that for tree-like networks the natural initial-boundary value problem admits semi-global classical solutions in the sense of Li [Li,T.T.,Controllability and Observability for Quasilinear Hyperbolic Systems,AIMS Ser.Appl.Math.,vol 3,American Institute of Mathematical Sciences and Higher Education Press,2010] existing in a neighborhood of the equilibrium solution.The authors then prove the local exact controllability of such networks near such equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.     

Markov network processes with product form stationary distributions

This study concerns the equilibrium behavior of a general class of Markov network processes that includes a variety of queueing networks and networks with interacting components or populations. The focus is on determining when these processes have product form stationary distributions. The approach is to relate the marginal distributions of the process to the stationary distributions of “node transition functions” that represent the nodes in isolation operating under certain fictitious environments. The main result gives necessary and sufficient conditions on the node transition functions for the network process to have a product form stationary distribution. This result yields a procedure for checking for a product form distribution and obtaining such a distribution when it exits. An important subclass of networks are those in which the node transition rates have Poisson arrival components. In this setting, we show that the network process has a product form distribution and is “biased locally balanced” if and only if the network is “quasi-reversible” and certain traffic equations are satisfied. Another subclass of networks are those with reversible routing. We weaken the known sufficient condition for such networks to be product form. We also discuss modeling issues related to queueing networks including time reversals and reversals of the roles of arrivals and departures. The study ends by describing how the results extend to networks with multi-class transitions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A simple system of reaction-diffusion equations describing morphogenesis: Asymptotic behavior

Summary A system of reaction-diffusion equations with a single non-linear term is investigated, such that the solutions remain bounded uniformly for infinite time, but the homogeneous equilibrium state is unstable. Using a Ljapunov functional and the compactness of the trajectories of the system, the solutions are proved to approach the equilibrium states. The stability character of the latter states is shown to be determined by a simpler selfadjoint problem. Entrata in Redazione il 29 giugno 1978.     

磁流体方程的时间解析性和时间周期解

考查了周期边界条件下的磁流体方程,证明了它的解关于时间是解析的,由此得到了磁流体方程的解的向后惟一性.对于周期解,证明了当周期小于某个常数时,周期的弱解是强解,进一步地这样的强解是定常解.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.     

On linear homogeneous almost periodic systems that satisfy the Favard condition

We prove the existence of a linear homogeneous almost periodic system of differential equations that has nontrivial bounded solutions and is such that all systems from a certain neighborhood of it have no nontrivial almost periodic solutions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3. pp. 409–413, March, 1998.     

On the dimension of the subspace of Liouvillian solutions of a Fuchsian system

The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described.     

Formation of singularities of solutions of the equations of motion of compressible fluids subjected to external forces in the case of several spatial variables

One considers a class of solutions with finite total energy and moment of inertia for the equations of motion of compressible fluids. It is shown that for a wide class of right-hand sides, including the viscosity term, initially smooth solutions may acquire singularities on a finite time interval. A sufficient condition for the appearance of singularities is found. This condition may be called “the best possible sufficient condition” in the sense that one can explicitly construct a time-global smooth solution for which this condition does not hold to within arbitrary infinitely small quantities. For a nontrivial constant state, perturbations with compact support are considered. A generalization is proved for the known theorem on the initial conditions for which the solution acquires singularities on a finite time interval. The effect of dry friction and rotation on the formation of singularities of smooth solutions is examined. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 274–308, 2007.     

Chaotic Dynamics of a Delayed Discrete-Time Hopfield Network of Two Nonidentical Neurons with no Self-Connections

A bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two delays, two internal decays and no self-connections, choosing the product of the interconnection coefficients as the characteristic parameter for the system. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified, and the existence of Fold/Cusp, Neimark–Sacker and Flip bifurcations is proved. All these bifurcations are analyzed by applying the center manifold theorem and the normal form theory. It is shown that the dynamics in a neighborhood of the null solution become more and more complex as the characteristic parameter grows in magnitude and passes through the bifurcation values. Under certain conditions, it is proved that if the magnitudes of the interconnection coefficients are large enough, the neural network exhibits Marotto’s chaotic behavior.      

On the asymptotic solutions of the equations of motion of mechanical systems

The existence of solutions of the equations of motion of non-natural mechanical systems of a certain form which tend to a position of equilibrium when the time increases without limit is proved by the methods described in /1, 2/. The corresponding instability theorem is proved.     

On immediate-delayed exchange of stabilities and periodic forced canards

Singularly perturbed nonautonomous ordinary differential equations are studied for which the associated equations have equilibrium states consisting of at least two intersecting curves, which leads to exchange of stabilities of these equilibria. The asymptotic method of differential equations is used to derive conditions under which initial value problems have solutions characterized by immediate and delayed exchange of stabilities. These results are then used to prove the existence of periodic canard solutions. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 1, pp. 46–61. The text was submitted by the authors in English.     

Regular and Irregular Solutions in the Problem of Dislocations in Solids

For an initial differential equation with deviations of the spatial variable, we consider asymptotic solutions with respect to the residual. All solutions are naturally divided into classes depending regularly and irregularly on the problem parameters. In different regions in a small neighborhood of the zero equilibrium state of the phase space, we construct special nonlinear distribution equations and systems of equations depending on continuous families of certain parameters. In particular, we show that solutions of the initial spatially one-dimensional equation can be described using solutions of special equations and systems of Schr¨odinger-type equations in a spatially two-dimensional argument range.     

On the convergence of solutions of certain inhomogeneous fourth-order differential equations

The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval.     

Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary

In this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction–diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter ? goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary.     

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane is studied. Under the entropy conditions, we obtain the solutions constructively. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a combustion wave CJDT into SDT in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution doesn’t contain combustion waves, which exhibits the instability for unburnt states. This work is supported by NSFC 10671120     

Modelling of dynamic networks of thin elastic plates

The purpose of this paper is to derive junction conditions for networks of thin elastic plates and to analyse the dynamic equations of such networks. Junction conditions for networks of Kirchhoff plates and networks of Reissner–Mindlin plates are derived based on geometric considerations of the deformation at a junction. It is proved that the dynamic system which describes the Reissner–Mindlin network is well-posed is an appropriate energy space. It is further established that the Kirchhoff network is obtained in the limit of the Reissner–Mindlin network as the shear moduli go to infinity.     

Stability of non-constant equilibrium solutions for two-fluid Euler–Maxwell systems

In this work we consider periodic problems for two-fluid compressible Euler–Maxwell systems for plasmas. The initial data are supposed to be in a neighborhood of non-constant equilibrium states. Mainly by an induction argument used in Peng (2015), we prove the global stability in the sense that smooth solutions exist globally in time and converge to the equilibrium states as the time goes to infinity. Moreover, we obtain the global stability of solutions with exponential decay in time near the equilibrium states for two-fluid compressible Euler–Poisson systems.     

Analysis of Local Dynamics of Difference and Close to Them Differential-Difference Equations

We study the local dynamics of one class of nonlinear difference equations which is important for applications. Using perturbation theory methods, we construct sets of singularly perturbed differential-difference equations that are close (in a sense) to initial difference equations. For the problem on the stability of the zero equilibrium state and for certain infinite-dimensional critical cases, we propose a method that allows us to construct analogs of normal forms. We mean special nonlinear boundary value problems without small parameters, whose nonlocal dynamics describes the structure of solutions to initial equations in a small neighborhood of the equilibrium state. We show that dynamic properties of difference and close to them differential-difference equations considerably differ.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.