Global Stability of a Rational Difference Equation

We consider the higher-order nonlinear difference equation +1=(+−)/(1++−),=0,1,… with the parameters, and the initial conditions −,…,0 are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002)

Stochastic Heavy Ball

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f. The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f. We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms

On Global Attractivity of a Class of Nonautonomous Difference Equations

We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given by y n p ry n−s / q φ n y n−1 , y n−2 , . . . , y n−m y n−s , n ∈ N 0 , with p ≥ 0, r, q > 0, s, m ∈ N and positive initial values, and present some sufficient conditions for the parameters and maps φ n : R m → R , n ∈ N 0 , under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted

Antiplane-inplane shear mode delamination between two second-order shear deformable composite plates

The second-order laminated plate theory is utilized in this work to analyze orthotropic composite plates with asymmetric delamination. First, a displacement field satisfying the system of exact kinematic conditions is presented by developing a double-plate system in the uncracked plate portion. The basic equations of linear elasticity and Hamilton’s principle are utilized to derive the system of equilibrium and governing equations. As an example, a delaminated simply supported plate is analyzed using Lévy plate formulation and the state-space model by varying the position of the delamination along the plate thickness. The displacements, strains, stresses and the J-integral are calculated by the plate theory solution and compared with those by linear finite-element calculations. The comparison of the numerical and analytical results shows that the second-order plate theory captures very well the mechanical fields. However, if the delamination is separated by only a relatively thin layer from the plate boundary surface, then the second-order plate theory approximates badly the stress resultants and so the mode-II and mode-III J-integrals and thus leads to erroneous results

Networks of Self-Adaptive Dynamical Systems

The present work belongs to the vast body of research devoted to behaviors that emerge when homogeneous or heterogeneous agents interact. We adopt a stylized point of view in which the individual agents' activities can be assimilated into nonlinear dynamical systems, each with their own set of specific parameters. Since the pioneering work of C. Huyghens in the seventeenth century it has been established that interactions between agents modify their individual evolutions – and that for ad-hoc interactions and agents that are not too dissimilar, synchronized behaviors emerge. In this classical approach, however, each agent recovers its individual evolution when interactions between them are removed or as summarized by a French aphorism:"Chasser le naturel et il revient au galop". The position we adopt in this work differs qualitatively from this classical approach. Here, we construct a mathematical framework that depicts the idea of systems interacting not only via their state variables, but also via a self-adaptive capability of the agents' local parameters. Specifically, we consider a network where each vertex is endowed with a dynamical system having initially different parameters. We explicitly construct adaptive mechanisms which, according to the system's state, tune the value of the local parameters. In our construction, the agents are modeled by dissipative ortho-gradient vector fields possessing local attractors (e.g. limit cycles). The forces describing the agents' interactions derive either from a generalized potential or from a linear combinations of coupling functions. Contrary to classical synchronization behavior which disappears when interactions are removed, here the system self-adapts and acquires consensual values for the set of local parameters. The consensual values are definitely "learned" (i.e. they stay in consensus even when interactions are removed). We analytically show for a wide class of dynamical systems how such a "plastic" and self-adaptive training of parameters can be achieved. We calculate the resulting consensual state and their relevant stability issues. The connectivity of the network (i.e. Fiedler number) affects the convergence rate but not the asymptotic consensual values. We then extend this idea to enable adaptation of parameters characterizing the coupling functions themselves. Self-learning mechanisms simultaneously operate at the agents' level and at the level of their connections. Finally, we analytically explore a set of dynamical systems involving the simultaneous action of two time-dependent networks (i.e. where edges evolve with time). The first network describes the interactions between the state variables, and the second affects the adaptive mechanisms themselves. In this last case, we show that for ad hoc time-dependent networks, parametric resonance phenomena occur in the dynamics. While our work puts a strong effort into explicit derivations and analytic results, we do not refrain from reporting a set of numerical investigations that show how our explicit construction can be implemented in various classes of dynamical systems

Stirring and mixing : 1999 Program of Summer Study in Geophysical Fluid Dynamics

The central theme of the 1999 GFD Program was the stirring, transport, reaction and mixing of passive and active tracers in turbulent, stratified, rotating fluids. The problem of mixing in fluids has applications in areas ranging from oceanography to engineering and astrophysics. In geophysical settings, mixing spans and unites a broad range of scales -- from micrometers to megameters. The mixing of passive tracers is of fundamental importance in environmental and industrial problems, such as pollution, and in determining the large-scale heat and salt balance of the worlds oceans. The transport of active tracers, on the other hand, such as vorticity, plays a key role in the turbulence that occurs in most geophysical and astrophysical fluids. William R. Young (Scripps Institution of Oceanography) gave a series of principal lectures, the notes of which as taken by the fellows, appear in this volume. Report of the projects of the student fellows makes up the second half of this volume.Funding was provided by the National Science Foundation under Grant No. OCE-9810647 and the Office of Naval Research under Grant No. NOO0l4-97-1-0934