# Volume 60, № 3, 2008

### On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 283–292

We consider the operator which is a variable hysteron that describes, according to the Krasnosel'skii -Pokrovskii scheme, a nonstationary hysteresis nonlinearity with characteristics varying under external influences.
We obtain sufficient conditions under which this operator is defined for inputs from the class of functions *H*^{1}[*t*_{0}, *T*]
that satisfy the Lipschitz condition on the interval [*t*_{0}, *T*].

### Asymptotic representations of the solutions of essentially nonlinear nonautonomous second-order differential equations

Belozerova M. A., Evtukhov V. M.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 310–331

We establish asymptotic representations for the solutions of a class of nonlinear nonautonomous second-order differential equations.

### Investigation of the periodic solutions of nonlinear autonomous systems in the critical case

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 332–339

We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case.

### Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 340–381

For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given.

### Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 382–387

Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation is constructed and its description is given in terms of covariant derivatives and the curvature form of the corresponding connection.

### Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 388–397

We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the Korteweg - de Vries equation with a small parameter at the higher derivative.

### Generalization of the Mukhamadiev theorem on the invertibility of functional operators in the space of bounded functions

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 398–412

We obtain necessary and sufficient conditions of reversibility of the linear bounded operator *d ^{m }* /

*d t*+

^{m }*A*in the space of functions bounded on

*R*.

### Periodic boundary-value problem for third-order linear functional differential equations

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 413–425

For the linear functional differential equation of the third order

*u''' *(*t*) = *l*(*u*)(*t*) + *q*(*t*),

theorems on the existence and uniqueness of a solution satisfying the conditions

*u*^{( i)}(0) = *u*^{( i)}, *i*=0,1,2,

are established. Here, *l* is a linear continuous operator transforming the space *C*([0, ω];*R*)
into the space *L*([0, ω];*R*), and *q* ∈ *L*([0, ω];*R*).
The question on the nonnegativity of a solution of the considered boundary-value problem is also studied.

### Amplitude synchronization in a system of two coupled semiconductor lasers

Lykova O. B., Schneider K. R., Yanchuk S. V.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 426–435

We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.