Analysis of a second-order hybrid system with linear structure

The paper is devoted to the investigation of the behavior of trajectories of a hybrid system on a plane consisting of a set of two-dimensional systems of linear differential equations and a discrete component—a rule for jumping from one system to another one when the trajectory of the system reaches the boundary of the domain of its consideration. The notion of quasiperiodic and asymptotically quasiperiodic trajectory is introduced; the structure of the set of initial positions that quasiperiodic or asymptotically quasiperiodic trajectories can be issued from is investigated. The Zeno behavior (an infinite number of jumps between systems of differential equations in a finite amount of time) is studied; sufficient conditions for a trajectory not to exhibit this type of behavior are formulated.     

Finite-Gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is “unpinched” to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.     

Study of a discrete dynamic system in a neighborhood of a quasi-periodic trajectory

We study a discrete system in a neighborhood of a quasi-periodic trajectory. We obtain conditions for reducing a system in this neighborhood to a system with quasi-periodic coefficients. We determine the behavior of this system under the action of small perturbations.This work was prepared with the financial support of the Ukrainian State Committee on Science and Technology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1702–1711, December, 1992.     

Quasiperiodicity,strange non-chaotic and chaotic attractors in a forced two degrees-of-freedom system

Strange non-chaotic, strange chaotic and quasiperiodic attractors are demonstrated to exist for a system of two non-linear coupled oscillators with almost periodic excitations. For same parameter values a transition from a strange non-chaotic to a quasiperiodic attractor is presented, whereas for other parameter values a shift from the strange chaotic attractor to a quasiperiodic one is found.     

Investigation of a discrete dynamical system in a neighborhood of the invariant torus

We study the behavior of a discrete dynamical system in a neighborhood of the invariant torus for the case where the trajectories may have arbitrary structure on the torus and establish conditions under which the system can be reduced to the canonical form in the indicated neighborhood.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1676–1685, December, 1995.     

Quasiperiodic solutions of degenerate linear systems of second-order ordinary differential equations

We establish sufficient conditions for the existence of quasiperiodic solutions of a system of ordinary second-order differential equations with degenerate symmetric matrix of the second derivatives for an arbitrary quasiperiodic inhomogeneity.     

A criterion of exponential dichotomy for a countable system of differential equations with quasiperiodic coefficients

We establish necesary and sufficient conditions for the exponential diehetemy with matrix projeseters of a countable system of differential equations with quasiperiodic coefficients.     

Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems

Quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems are studied. The behavior of solutions in a neighborhood of resonance and nonresonance levels is considered. Conditions for the existence of resonant quasiperiodic solutions (m-dimensional resonance tori) are found, and the global behavior of solutions in domains separated from the unperturbed separatrices is discussed. The results are illustrated by the example of the Duffing equation with the homoclinic figure eight of a saddle.     

Splitting and the spectrum of a linear differential equation with quasiperiodic coefficients

A linear differential equation with quasiperiodic coefficients for large values of the spectral parameter is split into an exponentially dichotomous system and a two-dimensional] system with the properties of the one-dimensional Schrödinger equation. The set of the values of the parameter, for which the equation has solution of the form A(t) epx (iat) with a real number a and a quasiperiodic function A(t), is a single out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1383–1388, October, 1990.     

On uniformly stable linear quasiperiodic systems

In a finite-dimensional complex space, we consider a system of linear differential equations with quasiperiodic skew-Hermitian matrix. The space of solutions of this system is a sum of one-dimensional invariant subspaces. Over a torus defined by a quasiperiodic matrix of the system, we investigate the corresponding one-dimensional invariant bundles (nontrivial in the general case). We find conditions under which these bundles are trivial and the system can be reduced to diagonal form by means of the Lyapunov quasiperiodic transformation with a frequency module coinciding with the frequency module of the matrix of the system.     

Local steering problem for a class of control-affine systems with application to continuous crystallization processes

In this paper, a nonlinear control-affine system that describes the dynamics of a continuous crystallizer with fines trap is considered. We study the approximate steering problem for this system, i. e. the problem of constructing an admissible open-loop control that steers the system from a given initial state to a neighborhood of the target state. This problem is reduced to solving a system of algebraic equations with respect to the control parameters. We show that the resulting algebraic system is locally solvable in a neighborhood of the equilibrium point. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conditions for the existence of a solution of a system of differential equations close to an approximate solution

We consider nonautonomous systems of differential equations and state conditions for the existence of an exact solution in a neighborhood of an approximate one by analyzing the linear system of the first approximation in a neighborhood of the constructed approximate solution. We present conditions for the existence of a bounded solution of a linear inhomogeneous system of differential equations.     

Super quasiperiodic wave solutions and asymptotic analysis for $$ \mathcal{N} = 1 $$ supersymmetric KdV-type equations

Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $ \mathcal{N} = 1 $ \mathcal{N} = 1 supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an N-periodic wave solution with arbitrary parameters for N ≥ 2. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.     

Parametric control of solutions to an evolution problem in a neighborhood of an unstable stationary regime

We study a controlled system of ordinary differential equations in a neighborhood of an unstable stationary regime. We seek for a control under which the solution remains in the neighborhood however long. We find conditions under which such control is possible and prove an existence theorem. The results are of a constructive character and can be applied to controlling actual processes.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Application of the GALI method to localization dynamics in nonlinear systems

We investigate localization phenomena and stability properties of quasiperiodic oscillations in N$N$ degree of freedom Hamiltonian systems and N$N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of N$N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date.     

Groups of <Emphasis Type="Italic">S</Emphasis>-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form \(\sqrt f /d{h^s}\), where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element \(\sqrt f \) is periodic. We also analyze the continued fraction expansion of the key element \(\sqrt f /{h^{g + 1}}\), which defines the set of quasiperiodic elements of a hyperelliptic field.     

Global solutions and invariant tori of difference-differential equations

We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.     

Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 525–528, March, 2006.     

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.