 EDITORINCHIEF
 DEPUTY EDITORINCHIEF
 Editorial board

Volume 23, Number 3
Carvalho M., Rodrigues A. P.
Complete Set of Invariants for a Bykov Attractor
Abstract
In this paper we consider an attracting heteroclinic cycle made by a 1dimensional and a 2dimensional separatrices between two hyperbolic saddles having complex eigenvalues. The basin of the global attractor exhibits historic behavior and, from the asymptotic properties
of these nonconverging time averages, we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. These invariants are determined by the quotient of the real parts of the eigenvalues of the equilibria, a linear combination of their imaginary
components and also the transition maps between two cross sections on the separatrices.

Ureña A. J.
The Spectrum of Reversible Minimizers
Abstract
Poincaré and, later on, Carathéodory, showed that the Floquet multipliers of 1dimensional periodic curves minimizing the Lagrangian action are real and positive. Even though Carathéodory himself observed that this result loses its validity in the general higherdimensional case, we shall show that it remains true for systems which are reversible in time. In this way, we also generalize a previous result by Offin on the hyperbolicity of nondegenerate symmetric minimizers. Our arguments rely on the higherdimensional generalizations of the Sturm theory which were developed during the second half of the twentieth century by several authors, including Hartman, Morse or Arnol’d.

Garashchuk I. R., Sinelshchikov D. I., Kudryashov N. A.
Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall
Abstract
Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.

Gelfreikh N. G.
Normal Forms for Threeparameter Families of Areapreserving Maps near an Elliptic Fixed Point
Abstract
We study dynamics of areapreserving maps in a neighborhood of an elliptic fixed point. We describe simplified normal forms for a fixed point of codimension 3. We also construct normal forms for a generic threeparameter family which contains such degeneracy and use
normal form theory to describe generic bifurcations of periodic orbits in these families.

Motonaga S., Yagasaki K.
Nonintegrability of Parametrically Forced Nonlinear Oscillators
Abstract
We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by secondorder homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and
Duffing oscillators to demonstrate our results.

Kim S.
On a Convex Embedding of the Euler Problem of Two Fixed Centers
Abstract
In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doublycovered elliptic coordinates provide a 2to1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have equal mass, but does not hold near the heavier body.

Bukh A. V., Slepnev A. V., Anishchenko V. S., Vadivasova T. E.
Stability and Noiseinduced Transitions in an Ensemble of Nonlocally Coupled Chaotic Maps
Abstract
The influence of noise on chimera states arising in ensembles of nonlocally coupled chaotic maps is studied. There are two types of chimera structures that can be obtained in such ensembles: phase and amplitude chimera states. In this work, a series of numerical experiments is carried out to uncover the impact of noise on both types of chimeras. The noise influence on a chimera state in the regime of periodic dynamics results in the transition to chaotic dynamics. At the same time, the transformation of incoherence clusters of the phase chimera to incoherence clusters of the amplitude chimera occurs. Moreover, it is established that the noise impact may
result in the appearance of a cluster with incoherent behavior in the middle of a coherence cluster.

Borisov A. V., Kilin A. A., Mamaev I. S.
A Nonholonomic Model of the Paul Trap
Abstract
In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the
hyperbolic paraboloid is made. A threedimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.
