We consider G=Γ×S1 with Γ being a finite group, for which the complete Euler ring structure in U(G) is described. The multiplication tables for Γ=D6, S4 and A5 are provided in the Appendix. The equivariant degree for G-orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the G-orthogonal degree extends the degree for G-gradient maps (in the case of G=Γ×S1) introduced by Gȩba in [K. Gȩba, W. Krawcewicz, J. Wu, An equivariant degree with applications ...
Tópico(s): Nonlinear Partial Differential Equations
2007 - Elsevier BV | Nonlinear Analysis
Tópico(s): Numerical methods in inverse problems
2005 - Birkhäuser | Milan Journal of Mathematics
Justyna Fura, Anna Ratajczak, Sławomir Rybicki,
In this article, we study the existence and the continuation of periodic solutions of autonomous Newtonian systems. To prove the results we apply the infinite-dimensional version of the degree for SO(2)-equivariant gradient operators defined by the third author in Nonlinear Anal. Theory Methods Appl. 23(1) (1994) 83–102 and developed in Topol. Meth. Nonlinear Anal. 9(2) (1997) 383–417. Using the results due to Rabier [Symmetries, Topological degree and a Theorem of Z.Q. Wang, J. Math. 24(3) (1994) 1087– ...
Tópico(s): Nonlinear Differential Equations Analysis
2005 - Elsevier BV | Journal of Differential Equations
E. N. Dancer, Kazimierz Gęba, Sławomir Rybicki,
Let $V$ be an orthogonal representation of a compact Lie group $G$ and let $S(V),D(V)$ be the unit sphere and disc of $V,$ respectively. If $F : V \rightarrow \mathbb R$ is a $G$-invariant $C^1$-map then the $G$-equivariant gradient $C^0$-map $\nabla F :
Tópico(s): Geometry and complex manifolds
2005 - Polish Academy of Sciences | Fundamenta Mathematicae
Wiktor Radzki, Sławomir Rybicki,
We study connected branches of nonconstant 2 π -periodic solutions of the Hamilton equation x ̇ (t)=λJ ∇ H(x(t)), where λ ∈(0,+∞), H∈C 2 ( R n × R n , R ) and ∇ 2 H(x 0 )= A 0 0 B for x 0 ∈∇ H −1 (0). The Hessian ∇ 2 H ( x 0 ) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given ( x 0 , λ 0 ). As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system x ̇ (t)=J ∇ H(x(t)) emanating from ...
Tópico(s): Nonlinear Differential Equations Analysis
2004 - Elsevier BV | Journal of Differential Equations
Andrzej J. Maciejewski, Sławomir Rybicki,
Tópico(s): Nonlinear Waves and Solitons
2004 - Springer Science+Business Media | Celestial Mechanics and Dynamical Astronomy
Norimichi Hirano, Sławomir Rybicki,
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).
Tópico(s): Control and Dynamics of Mobile Robots
2003 - Elsevier BV | Journal of Differential Equations
E. N. Dancer, Sławomir Rybicki,
1. Introduction.The aim of this paper is to formulate su cient conditions for the existence of global bifurcations of nonstationary periodic solutions of autonomous Hamiltonian systems.There are many bifurcation theorems describing bifurcations from non-degenerate critical points of Hamiltonians, see for example [6-9], [11], [12].For a discussion concerning bifurcations of periodic solutions of Hamiltonian systems we refer the reader to [19].On the other hand there are few results concerning the degenerate ...
Tópico(s): Nonlinear Waves and Solitons
1999 - Khayyam Publishing | Differential and Integral Equations
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1997 - Juliusz Schauder University Center for Nonlinear Studies | Topological Methods in Nonlinear Analysis
Let $\Lambda$ be the Laplace--Beltrami operator on $S^{n-1}$. The aim of this paper is to prove that any continuum of nontrivial solutions of the equation $-\Lambda u = f(u,\lambda),$ which bifurcate from the set of trivial solutions, is unbounded in $H^1(S^{n-1}) \times R$. As the main tool we use degree theory for $S^1$--equivariant, gradient operators defined in [15].
Tópico(s): advanced mathematical theories
1996 - Khayyam Publishing | Differential and Integral Equations
Tópico(s): Quantum chaos and dynamical systems
1994 - Elsevier BV | Nonlinear Analysis
Zalman Balanov, Wiesław Krawcewicz, Sławomir Rybicki, Heinrich Steinlein,
Tópico(s): Geometric and Algebraic Topology
2010 - Birkhäuser | Journal of Fixed Point Theory and Applications
Anna Gołębiewska, Sławomir Rybicki,
Tópico(s): Advanced Mathematical Modeling in Engineering
2010 - Elsevier BV | Nonlinear Analysis
Gabriel López Garza, Sławomir Rybicki,
We consider a bifurcation index BIFG(νk0−1)∈U(G) defined in terms of the degree for G-equivariant gradient maps, see Gȩba (1997) [21], Rybicki (1994) [22], Rybicki (2005) [23], where G is a real, compact, connected Lie group and U(G) is the Euler ring of G, see tom Dieck (1977) [29], tom Dieck (1987) [30]. The main result of this article is the following: BIFG(νk0−1)≠Θ∈U(G) iff BIFT(νk0−1)≠Θ∈U(T), where T⊂G is a maximal torus of G. It is also shown that all the bifurcation points of weak solutions of the ...
Tópico(s): Geometry and complex manifolds
2010 - Elsevier BV | Nonlinear Analysis
Ernesto Pérez-Chavela, Sławomir Rybicki, Daniel Strzelecki,
In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider a system $$(*)\; \ddot{q}= -\nabla U(q)$$ , where U(q) is a $$\Gamma $$ -invariant potential and $$\Gamma $$ is a compact Lie group acting linearly on $${\mathbb {R}}^n$$ . If system $$(*)$$ possess a non-degenerate orbit of stationary solutions $$\Gamma (q_0)$$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $$\nabla ^ ...
Tópico(s): Nonlinear Waves and Solitons
2017 - Springer Science+Business Media | Calculus of Variations and Partial Differential Equations
Anna Gołębiewska, Sławomir Rybicki,
In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.
Tópico(s): Nonlinear Waves and Solitons
2012 - American Institute of Mathematical Sciences | Discrete and Continuous Dynamical Systems - S
Andrzej J. Maciejewski, Wiktor Radzki, Sławomir Rybicki,
Tópico(s): Quantum Mechanics and Non-Hermitian Physics
2005 - Springer Science+Business Media | Journal of Dynamics and Differential Equations
Tópico(s): Advanced Mathematical Modeling in Engineering
1998 - Elsevier BV | Nonlinear Analysis
Kazimierz Gęba, Sławomir Rybicki,
Tópico(s): Algebraic Geometry and Number Theory
2008 - Birkhäuser | Journal of Fixed Point Theory and Applications
Justyna Fura, Sławomir Rybicki,
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 ...
Tópico(s): Differential Equations and Numerical Methods
2006 - Elsevier BV | Annales de l Institut Henri Poincaré C Analyse Non Linéaire
The aim of this article is to prove global bifurcation theorems forS1-equivariant potential operators of the form “compact perturbation of identity.” As an application we prove that components of the set of nontrivial solutions of system[formula]which bifurcate from the set of trivial solutions are unbounded in suitably chosen space. The important point to note here is that our results apply only to systems with an even number of equations. As the main tool we use degree theory forS1-equivariant ...
Tópico(s): Nonlinear Differential Equations Analysis
1998 - Elsevier BV | Journal of Mathematical Analysis and Applications
Marek Izydorek, Sławomir Rybicki,
Tópico(s): Algebraic Geometry and Number Theory
1992 - Springer Nature | Lecture notes in mathematics
Joanna Gawrycka, Sławomir Rybicki,
We study global bifurcation of weak solutions of systems of elliptic differential equations considered on SO(2)-invariant domains. We formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones. As the main tool we use the degree for SO(2)-equivariant gradient maps defined by the second author in Rybicki (Nonlinear Anal. TMA 23(1) (1994) 83).
Tópico(s): Stability and Controllability of Differential Equations
2004 - Elsevier BV | Nonlinear Analysis
Tópico(s): Advanced Operator Algebra Research
2001 - Elsevier BV | Nonlinear Analysis
Sławomir Rybicki, Piotr Stefaniak,
The aim of this paper is to show that any continuum of nontrivial solutions of a non-cooperative system of elliptic equations on the sphere S n − 1 , bifurcating from the set of trivial solutions, is unbounded. Moreover, we characterize bifurcation points of this system at which the global symmetry-breaking phenomenon occurs. As the main tool we use the degree for SO ( 2 ) -invariant strongly indefinite functionals defined in [13] .
Tópico(s): Stability and Controllability of Differential Equations
2015 - Elsevier BV | Journal of Differential Equations
Marek Izydorek, Sławomir Rybicki, Zbigniew Szafraniec,
The local scheme for an equilibrium state of an analytic planar dynamical systems is investigated.Upper bounds of the numbers of elliptic and hyperbolic sectors are derived.Methods of singularity theory are applied to obtain appropriate estimations in terms of indices of maps explicitly constructed from a vector field. I. IntroductionThe study of geometric differential equations was founded by H. Poincare in his classical "Memoire" [PCR1] (see alsoAt 15 years distance, Poincare's ideas was followed ...
Tópico(s): Geometric Analysis and Curvature Flows
1996 - Tokyo Institute of Technology | Kodai Mathematical Journal
Sławomir Rybicki, Naoki Shioji, Piotr Stefaniak,
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 ...
Tópico(s): Advanced Differential Equations and Dynamical Systems
2018 - De Gruyter | Advanced Nonlinear Studies
Ernesto Pérez-Chavela, Sławomir Rybicki, Daniel Strzelecki,
Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of Γ-symmetric systems q¨(t)=−∇U(q(t)) in any neighborhood of an isolated orbit of minima Γ(q0) of the potential U. We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.
Tópico(s): Astro and Planetary Science
2018 - Elsevier BV | Journal of Differential Equations
Zbigniew Błaszczyk, Anna Gołębiewska, Sławomir Rybicki,
We prove a version of the Poincare–Hopf theorem suitable for strongly indefinite functionals and then apply it to infer a number of bifurcation results in infinite-dimensional Hilbert spaces.
Tópico(s): Nonlinear Dynamics and Pattern Formation
2017 - | Advances in Differential Equations
Ernesto Pérez-Chavela, Sławomir Rybicki,
In this article we study topological bifurcations of classes of central configurations of the spatial 6- and 7-body problems. We treat these classes as SO(3)-orbits of critical points of a family of SO(3)-invariant potentials. Using the equivariant bifurcation theory technique, we prove the existence of a global topological bifurcation of classes of central configurations in the 7-body problem and a local topological bifurcation in the 6-body problem.
Tópico(s): Advanced Differential Equations and Dynamical Systems
2012 - Elsevier BV | Nonlinear Analysis Real World Applications