Oleg Zubelevich

Hill’s Formula

In 1886, in his study of stability of the lunar orbit, Hill (Acta Math. VIII(1):1–36, 1886) published a formula which expresses the characteristic polynomial of the monodromy matrix for a second order time periodic equation in terms of the determinant of a certain infinite matrix. Here we present several versions of this formula and give its applications in the problem of dynamical stability.

The Anti-Integrable Limit

Anti-integrable limit is one of convenient and relatively simple methods for construction of chaotic hyperbolic invariant sets in Lagrangian, Hamiltonian and other dynamical systems. We discuss the most natural context of the method -- discrete Lagrangian systems. Then we present examples and applications.

The Continuous Averaging Method

There are several problems in perturbation theory, where standard methods do not lead to satisfactory results. We mention as examples the problem of an inclusion of a diffeomorphism into a flow in the analytic set up, and the problem of quantitative description of exponentially small effects in dynamical systems. In these cases one possible approach is based on the continuous averaging. The method appeared as an extension of the Neishtadt averaging procedure (Neishtadt in Prikl. Mat. Meh. 46(2):197–204, 1984) effectively working in the presence of exponentially small effects.

Abstract version of the Cauchy-Kowalewski problem

We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.

A fixed point theorem for affine mappings and its application to elasticity theory

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.

The Separatrix Map

The separatrix map was invented to study dynamical systems near asymptotic manifolds. It was introduced by Zaslavsky and Filonenko (Zh. Eksp. Teor. Fiz. 54:1590–1602, 1968) (see also Chirikov in Phys. Rep. 52(5):263–379, 1979; Filonenko et al. in Nucl. Fus. 7:253–266, 1967) for near-integrable Hamiltonian systems with one-and-a-half degrees of freedom and independently by Shilnikov (Sov. Math. Dokl. 6:163–166, 1965) in generic systems. In this chapter we obtain explicit formulas for the Zaslavsky separatrix map. These results are used in Chap. 5 for studying of the dynamics in the stochastic layer.

Splitting of Asymptotic Manifolds

The behavior of manifolds which are asymptotic to equilibriums, to periodic solutions, or in general to hyperbolic tori determines many features of chaos in dynamical systems. In this chapter we present the Poincare–Melnikov theory of splitting of asymptotic manifolds (separatrices) in Hamiltonian systems with one and a half degrees of freedom and in two-dimensional symplectic maps. Then we discuss the multidimensional version of this theory.

Introduction to the KAM Theory

The Kolmogorov–Arnold–Moser theory showed that quasi-periodic motions are generic in Hamiltonian systems. Moreover, they usually form a set of a positive measure in the phase space. This changed considerably the generally accepted idea of the dynamics in Hamiltonian systems close to integrable. Earlier such systems were supposed to be as a rule ergodic on compact energy levels (A dynamical system is called ergodic with respect to an invariant probability measure on the phase space if the measure of any invariant set equals zero or one.). In the present chapter we discuss basic facts and ideas of the KAM theory and prove one of the simplest theorems of this type.

Width of the Stochastic Layer

In this chapter we use the separatrix map to study the stochastic layer, appearing in the vicinity of separatrices of near-integrable systems.