ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.

DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

In the given article for the mixed-type equation with a singular coefficient the first boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justification of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

Keldysh problem for Pulkin’s equation in a rectangular domain

In this article for the mixed type equation with a singular coefficient Keldysh problem of incomplete boundary conditions is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral prob- lem the criterion of uniqueness is established. The solution is constructed as the summary of Fourier-Bessel row. At the foundation of the uniform convergence of a row there is a problem of small denominators.Under some restrictions on these tasks evaluation of separation from zero of a small denominator with the corresponding asymptotics was found, which helped to prove the uniform con- vergence and its derivatives up to the second order inclusive, and the existence theorem in the class of regular solutions.

The Rossby wave extra invariant in the dynamics of 3-D fluid layers and the generation of zonal jets

We consider an adiabatic-type (approximate) invariant that was earlier obtained for the quasi-geostrophic equation and the shallow water system; it is an extra invariant, in addition to the standard ones (energy, enstrophy, momentum), and it is based on the Rossby waves. The presence of this invariant implies the energy transfer from small-scale eddies to large-scale zonal jets. We show that this extra invariant can be extended to the dynamics of a three-dimensional (3-D) fluid layer on the beta plane. Combined with the investigation of other researchers, this 3-D extension implies enhanced generation of zonal jets. For a general physical system, the presence of an extra invariant (in addition to the energy–momentum and wave action) is extremely rare. We summarize the unique conservation properties of geophysical fluid dynamics (with the beta effect) that allow for the existence of the extra invariant, and argue that its presence in various geophysical systems is a strong indication that the formation of zonal jets is indeed related to the extra invariant. Also, we develop a new, more direct, way to establish extra invariants (without using cubic corrections). For this, we introduce the small denominator lemma.

Hilbert series and applications to graded rings

This paper contains a number of practical remarks on Hilbert series that we expect to be useful in various contexts. We use the fractional Riemann-Roch formula of Fletcher and Reid to write out explicit formulas for the Hilbert seriesP(t)in a number of cases of interest for singular surfaces (see Lemma 2.1) and3-folds. IfXis aℚ-Fano3-fold andS∈ |−KX|aK3surface in its anticanonical system (or the general elephant ofX), polarised withD=𝒪S (−KX), we determine the relation betweenPX(t)andPS,D(t). We discuss the denominator∏(1−tai)ofP(t)and, in particular, the question of how to choose a reasonably small denominator. This idea has applications to findingK3surfaces and Fano3-folds whose corresponding graded rings have small codimension. Most of the information about the anticanonical ring of a Fano3-fold orK3surface is contained in its Hilbert series. We believe that, by using information on Hilbert series, the classification ofℚ-Fano3-folds is too close. FindingK3surfaces are important because they occur as the general elephant of aℚ-Fano 3-fold.

The double Hopf bifurcation with 2:1 resonance

Many problems of physical interest involve the nonlinear interaction of two oscillators with different frequencies. When these frequencies are incommensurate, the interaction is straightforward, involving the amplitude rather than the phase of each oscillator. When the frequencies are in a ratio closely corresponding to a rational fraction of small denominator, phase locking occurs and the dynamics is much richer. In this paper we investigate fully the important case where the frequencies are in the ratio 2:1. The analysis allows for small departures from this ratio and treats not only the generic problem for small amplitudes but also an important degenerate case which appears to capture most of the dynamics at larger amplitudes.