Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincaré map is an integer and the slow curve is connected, the number of canard limit cycles is bounded from above by the number of fold points of the slow curve. In the present paper, it is proved that there are no such geometric constraints for non-integer rotation numbers: it is possible to construct a generic system with “as simple as possible” slow curve and arbitrary many limit cycles.

We will consider the exact controllability of the distributed system, governed by string equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case, the memory kernel is a linear combination of exponentials.

We prove that simplest Morse-Smale systems can have locally flat and wildly embedded separatrices of saddle periodic point.

For a generic skew product with the fiber a circle over an Anosov diffeomorphism, we prove that the Milnor attractor coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence, we conclude that such skew product is either transitive or has non-wandering set of zero measure. The result is proved under the assumption that the fiber maps preserve the orientation of the circle, and the skew product is partially hyperbolic.

In the present paper we consider preserving orientation Morse-Smale diﬀeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diﬀeomorphisms have a ﬁnite number of orientable heteroclinic orbits.

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamentaleffect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B,A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B,A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber,

O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂Lr of the phase-lock area Lr with the line {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding “conjugate” double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M.Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock area Lr lie in its axis. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr . In this paper, we present new results partially confirming both conjectures. The main result states that for every r ∈ Z \ {0} the phase-lock area Lr contains the infinite interval of its axis issued upwards from the simple intersection in ∂Lr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both nonzero at each constriction.We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction.

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the partial classification of k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invisibility conjecture.

The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a poly- nomial solution. They have shown that for μ ≠ 0 this happens ex- actly when l ∈ N and the parameters (λ, μ) lie on an algebraic curve Γl ⊂ C2(λ,μ) called the l-spectral curve and defined as zero locus of de- terminant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dy- namical systems on 2-torus depending on real parameters (B,A;ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in R2(B,A) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every l ∈ N. We also calculate its genus for l ⩽ 20 and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in R3 . We show that, unexpectedly, its complexification is(B,A,ω−1) a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.

In this paper, we study the third Painlevé equation with parameters *γ* = 0, *α**δ* ≠ 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters *δ* = −*β^*2/2 ≠ 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.

Consider a sofic dynamical system. We obtain an explcit formula for the KS-entropy of sofic dynamicsl system of Blackwell's type.

We present a special kind of normalization theorem: linearization theorem for skew products. The normal form is a skew product again, with the fiber maps linear. It appears that even in the smooth case, the conjugacy is only Hölder continuous with respect to the base. The normalization theorem mentioned above may be applied to perturbations of skew products and to the study of new persistent properties of attractors.

Algorithm is given for computation of the Hausdorff dimension of the support of the Erdos measure for a Pisot number.