Zalman Balanov

Geometric methods in degree theory for equivariant maps

Fundamental domains and extension of equivariant maps.- Degree theory for equivariant maps of finite-dimensional manifolds: Topological actions.- Degree theory for equivariant maps of finite-dimensional manifolds: Smooth actions.- A winding number of equivariant vector fields in infinite dimensional banach spaces.- Some applications.

Symmetric systems of Van Der Pol equations

We study the impact ofnsymmetries on the occurrence of periodic solutions in systems of nvan der Pol equations. We apply the equivariantndegree theory to establish existence results for multiple nonconstant nperiodic solutions and classify their symmetries. The computations of the nalgebraic invariants in the case of dihedral, tetrahedral, octahedral and nicosahedral symmetries for a van der Pol system of equations are included.

Morse complex, even functionals and asymptotically linear differential equations with resonance at infinity

I. Motivation. Let H be a Hilbert space and f : H → R a C-functional. To study critical points of f in the framework of the classical approaches (Morse Theory [39], Ljusternik–Schnirelman theory [40], etc.) one needs to assume, in particular, that f satisfies the Palais–Smale condition (in short, PS-condition): any sequence {xn} ⊂ H with {f(xn)} bounded and ∇f(xn) → 0 contains a convergent subsequence. In turn, the PS-condition is closely related to deformation properties of the flows associated with gradient vector fields. At the same time, as is well-known, there are many important variational problems, where the corresponding functionals fail to satisfy the PS-condition in any suitable sense. In addition, these functionals may not satisfy certain other conditions that are necessary for application of the classical methods. The problem of weakening the PS-condition has attracted a considerable attention for a long time (see, for instance, [16], [20], [21], [49] and references therein). An essential step in this direction was done by C. Conley [21] who

A comparison principle and extension of equivariant maps

We introduce a notion of a fundamental domain for an action of a compact Lie group on an arbitrary metric space and use this concept in order to prove a rather general version of a comparison principle for degrees of equivariant maps. Some applications of these results are also discussed.

BUCKLING OF A THIN ELASTIC PLATE

Abstract The global solvability and the bifurcation phenomena for the von Karman equations are studied by means of the Morse complex technique. Homological properties of solutions are also discussed.

Noninvasive Stabilization of Periodic Orbits in O4-Symmetrically Coupled Systems Near a Hopf Bifurcation Point

Pyragas time-delayed feedback control has proven itself as an effective tool to noninvasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings. In this paper, we consider O4-symmetric systems of van der Pol and optical oscillators coupled in a cube-like configuration. These systems undergo equivariant Hopf bifurcations giving rise to multiple branches of unstable periodic solutions. We introduce a delayed control term, which ensures stabilization of a selected branch. Group theoretic restrictions which help to shape our choice of control are discussed. Furthermore, we explicitly describe the domains in a two-dimensional parameter space for which the periodic solutions of the delayed system are stable.

On the genus of some subsets of

The problem of estimating the genus of G-spaces (G-category) attracts considerable attention (see, for instance, [Bar1, Bar2, Fa, FaHu, Kr, LS, Šv] and others). At least two approaches to this problem exist: geometric, based on Borsuk–Ulam type theorems, and homological, based on (co)homological arguments in the study of orbit spaces. Historically, the first result concerning this problem is the famous Lusternik– Schnirelman Theorem stating that the category of the n-dimensional real projective space equals n + 1 (see [LS]). In terms of genus the Lusternik–Schnirelman Theorem can be formulated as follows: the genus of the n-dimensional sphere with respect to the antipodal action is equal to n + 1. This result was generalized by A. Fet [Fe] to the case of an arbitrary free involution on the sphere. The case of a free action of an arbitrary finite cyclic group was considered by M. Krasnosel’skĭı [Kr] in the framework of the geometric approach. A. Švartz [Šv] was the first to consider the case of a non-free action of a cyclic group on the sphere and obtained the following result: let the finite cyclic group Zp act on the n-dimensional unit sphere S, let A = {x ∈ S | ∃g ∈ Zp : g 6= 1 & gx = x}, and suppose dim A = k. Then gen(S A) ≥ n− k, where gen(·) denotes genus.

G

Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl depend on symmetries of coupling. In this paper, we consider

-spheres

N^3

Periodic solutions of vdP and vdP-like systems on 3-tori

identical vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the same way as for a