arXiv subjects
Publications and source records attributed to S. Rybicki.
The aim of this paper is to formulate necessary conditions and sufficient ones for the existence of closed connected sets of nonstationary $2 \pi$-periodic solutions of $S^1$-symmetric Newtonian systems in $C_{2 \pi}([0,2\pi],\Omega) \times (0,+ \infty)$. As the main topological tool we apply the degree for equivariant gradient maps.
In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions of some well-known theorems imply that of our main results. We obtain our results with the use of the theory of normal forms for Hamiltonian matrices and global bifurcation theory for autonomous Hamiltonian systems.
In this article we study the existence, continuation and bifurcation from infinity of nonconstant solutions for a nonlinear Neumann problem. We apply the Leray-Schauder degree and the degree for SO(2)-equivariant gradient operators defined by the second author.
In this paper we prove the existence of non-stationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the degree for $S^1$-equivariant maps.
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. Using the results due to Rabier we show that the Leray-Schauder degree is not applicable in the proofs of our theorems, because it vanishes.
We study connected branches of non-constant {$2π$-pe}riodic solutions of the Hamilton equation \begin{displaymath} \dot{x}(t)=λJ\nabla H(x(t)), \end{displaymath} where $λ\in\halfline,$ $H\in C^2(\R^n\times\R^n,\R)$ and $ \displaystyle \nabla^2H(x_0)= [ \begin{array}{cc} A&0 0&B \end{array} ] $ for $x_0\in\nabla H^{-1}(0).$ The Hessian $\nabla^2H(x_0)$ 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 $\dot{x}(t)=J\nabla H(x(t))$ emanating from $x_0.$ We describe also minimal periods of trajectories near $x_0.$
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. Using the results due to Rabier 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.