Nonlinear spatiotemporal stability of solitons against external perturbations

We study nonlinear stability of solitons under the action of external spatiotemporal perturbations. Previous results on stationary structures [Rizzato FB, de Oliveira GI, Chian AC-L. Nonlinear stability of solitons against strong external perturbations. Phys Rev E 2003;67:047601] are extended to show that the centroid of moving solitons may develop chaotic dynamics, which under certain circumstances affects the corresponding envelope dynamics of these modes and reduces their stability. Applications to plasmas and Bose–Einstein condensates are discussed.     

Nonlinear dynamics and solitons in spin glasses

Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 2, pp. 163–174, May, 1990     

Global dynamics of the periodic logistic system with periodic impulsive perturbations

This paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations. The results of D.D. Bainov and P.S. Simeonov (1993) are extended and dynamics different from the corresponding continuous system are found. It is shown that the system may have a unique positive periodic solution which is globally asymptotically stable, or go extinct when the two periods are rational dependent. When they are rational independent, the system has no periodic solutions, however, still has a global attractor or go extinct under some conditions.     

Exponential averaging for traveling wave solutions in rapidly varying periodic media

Reaction diffusion systems on cylindrical domains with terms that vary rapidly and periodically in the unbounded direction can be analyzed by averaging techniques. Here, using iterated normal form transformations and Gevrey regularity of bounded solutions, we prove a result on exponential averaging for such systems, i.e., we show that traveling wave solutions can be described by a spatially homogenous equation and exponentially small remainders. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of particle dynamics in a rapidly varying viscous flow

A method for the numerical simulation of the dynamics of particles in a rapidly varying viscous flow has been developed and implemented as a software package. The frequency of variations in the fluid velocity is assumed to be such that the nonlinear terms in the equations of motion can be neglected in comparison with the nonstationary terms. The hydrodynamic interaction of the particles is taken into account. The velocities of the particles and their trajectories are computed. It is found that the trajectories of the particles depend substantially on the ratio of their radii. In the case of dipole particles in a rapidly varying external magnetic field, the hydrodynamic interaction is shown to prevent the particles from approaching each other under the influence of dipole-dipole interaction forces.     

Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

We study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h^-n+1) for the number of the resonances of P(h) lying in a disk of radius h.     

Moving gap solitons in periodic potentials

We address the existence of moving gap solitons (traveling localized solutions) in the Gross–Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit solutions of the coupled‐mode system. We show, however, that exponentially decaying traveling solutions of the Gross–Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled‐mode formalism and are estimated by using techniques of spatial dynamics and local center‐stable manifold reductions. Existence of bounded traveling solutions of the Gross–Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Emergence of epidemics in rapidly varying networks

We describe a simple model mimicking disease spreading on a network with dynamically varying connections, and investigate the dynamical consequences of switching links in the network. Our central observation is that the disease cycles get more synchronized, indicating the onset of epidemics, as the underlying network changes more rapidly. This behavior is found for periodically switched links, as well as links that switch randomly in time. We find that the influence of changing links is more pronounced in networks where the nodes have lower degree, and the disease cycle has a longer infective stage. Further, when the switching of links is periodic we observe finer dynamical features, such as beating patterns in the emergent oscillations and resonant enhancement of synchronization, arising from the interplay between the time-scales of the connectivity changes and that of the epidemic outbreaks.     

Attitude dynamics of a spacecraft with variable structure at presence of harmonic perturbations

Attitude dynamics of a spacecraft (SC) with variable structure (inertia–mass parameters variation) is examined. Equations of the motion of the SC are obtained on the base of Hamiltonian formalism in Serret–Andoyer variables. These equations can be used for analysis and synthesis of conditions of the SC attitude motion on active legs of orbital trajectories. Analytical and numerical modeling of the SC motion is realized. Existence of the SC chaotic modes of motion is demonstrated with the help of Melnikov method and Poincaré sections. Also attitude motion of a dual-spin spacecraft (DSSC) is considered at presence of small internal harmonic torque between DSSC coaxial bodies.     

Some singular perturbations of periodic operators

We consider two examples of singular perturbations of a periodic differential operator on the axis. The first example is a rapidly oscillating potential with compact support, and the second example is the delta potential with a small complex coupling constant. We investigate the structure and asymptotic behavior of the spectrum of the perturbed operators in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 207–218, May, 2007.     

Some properties of rapidly varying sequences

We investigate the class R∞,s of rapidly varying sequences from different points of view. It is shown: (1) a sequence (cn)_n∈N of positive real numbers is rapidly varying if and only if the corresponding function x→c[x], x?1, is rapidly varying; (2) the class R∞,s satisfies some selection properties, as well as game-theoretical and Ramsey-theoretical conditions.     

Nonlinear perturbations of positive semigroups

Communicated by J. A. Goldstein     

Descent methods with linesearch in the presence of perturbations

We consider the class of descent algorithms for unconstrained optimization with an Armijo-type stepsize rule in the case when the gradient of the objective function is computed inexactly. An important novel feature in our theoretical analysis is that perturbations associated with the gradient are not assumed to be relatively small or to tend to zero in the limit (as a practical matter, we expect them to be reasonably small, so that a meaningful approximate solution can be obtained). This feature makes our analysis applicable to various difficult problems encounted in practice. We propose a modified Armijo-type rule for computing the stepsize which guarantees that the algorithm obtains a reasonable approximate solution. Furthermore, if perturbations are small relative to the size of the gradient, then our algorithm retains all the standard convergence properties of descent methods.     

Discrete spectrum in the gaps for perturbations of periodic Jacobi matrices

In the paper we study the problem of the finiteness of the discrete spectrum for operators generated by perturbations of periodic Jacobi matrices. In particular, estimate formulae for the number of the eigenvalues created by perturbations in the gaps of the unperturbed operator are established.