具有时变时滞惯性四元Hopfield神经网络反周期解的动力学研究

周期性、反周期性和概周期性是时变神经网络的重要动态行为特性.本文在不将所研究的神经网络分解为实值系统的情况下,根据重合度理论中的延拓定理和不等式技巧,通过构造不同于现有平衡点稳定性研究的李雅普诺夫函数,研究了一类具有变时滞的惯性四元Hopfield神经网络的反周期解的动力学问题,给出了上述神经网络反周期解存在的一个新的判别条件.并通过构造李雅普诺夫函数论证了上述神经网络反周期解的指数稳定性.     

Coupled-Mode Equations and Gap Solitons in a Two-Dimensional Nonlinear Elliptic Problem with a Separable Periodic Potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier–Bloch decomposition and the implicit function theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a nondegeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross–Pitaevskii equation approximated by solutions of the coupled-mode equations are obtained for a finite-time interval.     

Upper estimates for the number of periodic solutions to multi-dimensional systems

The maximal number of zeros of multi-dimensional real analytic maps with small parameter is studied by means of the multi-dimensional generalization of Rouché's theorem. The obtained result is applied to study the maximal number of periodic solutions to multi-dimensional differential systems. An application to a class of three-dimensional autonomous systems is given.     

On the investigation of oscillations of quasilinear systems with almost-periodic coefficients : PMM vol.43, no. 6, 1979, pp. 980–991

A method is developed for investigating the oscillations of systems with almost-periodic coefficients, based on Kamenkov's ideas [1] on the construction of stationary solutions of systems with periodic coefficients and on the separation of motions. In contrast to [1] it is assumed that under the vanishing of a small parameter μ the system's characteristic equation has, besides n pairs of pure imaginary roots, m zero roots and h roots with negative real parts. Non-resonance and resonance cases are considered. Conditions are obtained for the existence of stationary solutions with respect to terms of first order in the small parameter. An example is presented.     

Propagation failure in discrete periodic media

We study the periodic lattice dynamical systems with bistable nonlinearity. We use Moser's theorem to show that there exist infinitely many stationary solutions when one of the migration coefficients is sufficiently small. Moreover, we prove that the propagation failure occurs when both migration coefficients are sufficiently small.     

On the Floquet problem for second-order Marchaud differential systems

Solutions in a given set of the Floquet boundary value problem are investigated for second-order Marchaud systems. The methods used involve a fixed point index technique developed by ourselves earlier with a bound sets approach. Since the related bounding (Liapunov-like) functions are strictly localized on the boundaries of parameter sets of candidate solutions, some trajectories are allowed to escape from these sets. The main existence and localization theorem is illustrated by two examples for periodic and anti-periodic problems.     

Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

By using an approach developed by one of the authors, approximate solutions of the soft periodic boundary conditions for a two-cell reaction diffusion model have been obtained. The system is considered with reactant A and autocatalyst B. The reaction is taken cubic in the autocatalyst in the two-cell with linear exchange through A. The formal exact solution is obtained which is symmetric with respect to the mid-point of the container. Approximate solutions are found through the Picard iterative sequence of solutions constructed after the exact one. It is found that the solution obtained is not unique. When the initial conditions are periodic, the most dominant modes initiate to traveling waves in systems with moderate size. Symmetric configurations forming a parabolic one for large time are observed. In systems of large size, spatially symmetric chaos are produced which are stationary in time. Furthermore, it is found the symmetric pattern formation hold irrespective of the condition of linear instability against small spatial disturbance.     

Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as “periodic orbits up to an isometry” for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, without destroying the selfdual character of the system. N. Ghoussoub was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. A. Moameni’s research was supported by a postdoctoral fellowship at the University of British Columbia.     

A Coupled System of Caputo-Type Sequential Fractional Differential Equations with Coupled (Periodic/Anti-periodic Type) Boundary Conditions

In this paper, we introduce the concept of coupled (periodic/anti-periodic type) boundary conditions and solve a coupled system of nonlinear sequential fractional differential equations equipped with these conditions. Sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are presented. Our results are new in the given configuration and are well illustrated with the aid of examples.     

On positive periodic solutions of the time–space periodic Lotka–Volterra cooperating system in multi-dimensional media

This work is devoted to the study of the time–space periodic reaction–diffusion–advection Lotka–Volterra cooperating system in multi-dimensional media. By using the method of sub-super solutions and its associated iterations, we prove the existence and uniqueness of the positive periodic solution under appropriate conditions. Finally, we are able to derive the asymptotic behavior of the solutions to the associated Cauchy problem.     

Lyapunov-type stability criterion for periodic generalized Camassa–Holm equations

We study the Lyapunov stability of the periodic generalized Camassa–Holm equation in terms of the periodic/anti-periodic eigenvalues and the associated spectral intervals. Moreover, we establish a Lyapunov-type stability criterion based on the Floquet theory and a Lyapunov-type inequality.     

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.     

On the spectrum for the artificial compressible system

Stability of stationary solutions of the incompressible Navier–Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number ? which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small ?. The result is applied to the Taylor problem.     

一类时滞系统的周期解

本文通过Lyapunov第二方法给出了一类时滞系统周期解和平稳振荡存在性的方便实用的两个定理 ,并推广了一些已知的结论 .     

The Rotation Number Approach to Eigenvalues of the One-Dimensional p-Laplacian with Periodic Potentials

The paper studies the periodic and anti-periodic eigenvaluesof the one-dimensional p-Laplacian with a periodic potential.After a rotation number function () has been introduced, itis proved that for any non-negative integer n, the endpointsof the interval ^–1(n/2) in R yield the corresponding periodicor anti-periodic eigenvalues. However, as in the Dirichlet problemof the higher dimensional p-Laplacian, it remains open if theseeigenvalues represent all periodic and anti-periodic eigenvalues.The result obtained is a partial generalization of the spectrumtheory of the one-dimensional Schrödinger operators withperiodic potentials.     

Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.     

Conditional stability for periodic solutions of periodic systems containing a small parameter

We treat periodic solutions to periodic systems of ordinary differential equations, having a small parameter. Conditional stability and the dimensionality of the asymptotically stable manifolds are determined by the degree of stability of rest points of related iterated average systems. Finally, rates of decay for solutions on these manifolds are found, and for linear systems, the Floquet multipliers are determined.     

Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses

In this paper we investigate a class of Hopfield neural networks subject to periodic impulses. First we give sufficient conditions to ensure existence and exponential stability of the anti-periodic solutions, which are new and complementary to previously known results. Then we present an example to demonstrate our results.     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.     

Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: Analytical solution and collisions with application to Rossby waves

Coupled-Nonlinear Schrödinger equations, linked by cross modulation terms, arise in both nonlinear optics and in Rossby waves in the atmosphere and ocean. In this paper, we derive exact, analytic solutions for the “bright” coupled-mode soliton, for which the envelope in each mode asymptotes to zero at spatial infinity, and for its spatially periodic generalization. We then numerically study the collisions of the coupled-mode solitary wave both with a conventional envelope soliton, confined to a single mode, and also with a second coupled-mode solitary wave. The collisions are sensitive to both the relative speed and phase of the solitons. In some parameter ranges, the collisions are nearly elastic, but in others, one or both solitons fission.