Sławomir Rybicki

Existence of limit cycles for coupled van der Pol equations

Abstract In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S1-degree theory due to Dylawerski et al. (see Ann. Polon. Math. 62 (1991) 243).

Periodic solutions of second order Hamiltonian systems bifurcating from infinity

Abstract The goal of this article is to study closed connected sets of periodic solutions, of autonomous second order Hamiltonian systems, emanating from infinity. The main idea is to apply the degree for SO ( 2 ) -equivariant gradient operators defined by the second author in [S. Rybicki, SO ( 2 ) -degree for orthogonal maps and its applications to bifurcation theory, Nonlinear Anal. TMA 23 (1) (1994) 83–102]. Using the results due to Rabier [P. Rabier, Symmetries, topological degree and a theorem of Z.Q. Wang, Rocky Mountain J. Math. 24 (3) (1994) 1087–1115] we show that we cannot apply the Leray–Schauder degree to prove the main results of this article. It is worth pointing out that since we study connected sets of solutions, we also cannot use the Conley index technique and the Morse theory.

Global Bifurcations of Periodic solutions of the Hill Lunar Problem

We describe global bifurcations of non-stationary periodic solutions of the Hill Lunar Problem. Especially we are interested in description of closed connected sets (continua) of non-stationary periodic solutions which bifurcate from stationary ones. Such continua of solutions of the Hill Lunar Problem are not admissible in H12π \ Λ(H). For the Regularized Hill Lunar Problem we prove that these families are unbounded in H12π. As the main tool we use degree theory for SO(2)-equivariant orthogonal maps defined by S.M. Rybicki.

Solutions of Multiparameter Systems of Elliptic Differential Equations

Abstract In this article we study bifurcations of weak solutions of the following multiparameter variational system of elliptic differential equations: We formulate necessary and sufficient conditions for the existence of bifurcation points and global bifurcation points of weak solutions of this system.

Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

Abstract The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in S n {S^{n}} . In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO ⁡ ( n ) {\operatorname{SO}(n)} -symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO ⁡ ( n ) {\operatorname{SO}(n)} -invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

Some remarks on degree theory for

The aim of this article is to introduce a new class

{\text{\rm SO}(2)}

\text{\rm SO}(2)

-equivariant transversal maps

-equivariant transversal maps

Existence and continuation of periodic solutions of autonomous Newtonian systems

\text{\rm cl}(\Omega),\partial \Omega)

Classification of homotopy classes of equivariant gradient maps

and to define degree theory for such maps. We define degree for