Periodic solutions of symmetric autonomous Newtonian systems

In this paper we show the existence and bifurcation of T-periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space R2n is equipped with the structure of an orthogonal representation (W,ρW) and the potential is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on W only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period.     

From graphs to ortholattices and equivariant maps

A functor is constructed from the category of graphs and graph homomorphisms to the category of spaces with involutions and equivariant homotopy classes of maps. This can sometimes be used to prove lower bounds on chromatic numbers, and was inspired by Lovász's proof of Kneser's conjecture. Ortholattices occur as an intermediate step between graphs and spaces, and the correspondence between graphs and ortholattices is analyzed.     

Otopy classes of equivariant maps

The aim of this work is to study otopy classes of equivariant local maps. We present some extensions of the topological degree to equivariant local maps both in the gradient and non-gradient case and explain the relation between these two generalizations.     

Degree of equivariant maps between generalized G-manifolds

Yasuhiro Hara in [Topology Appl. 148 (2005), 113–121] and Jan Jaworowski in [J. Fixed Point Theory Appl. 1 (2007), 111–121] studied, under certain conditions, the degree of equivariant maps between free G-manifolds, where G is a compact Lie group. The main results obtained by them involve data provided by the classifying maps of the orbit spaces. In this paper, we extend these results by replacing the free G-manifolds by free generalized G-manifolds over ${\mathbb{Z}}$ .     

Conjugate gradient and minimal residual methods for solving symmetric indefinite systems

Norm-minimizing-type methods for solving large sparse linear systems with symmetric and indefinite coefficient matrices are considered. The Krylov subspace can be generated by either the Lanczos approach, such as the methods MINRES, GMRES and QMR, or by a conjugate-gradient approach. Here, we propose an algorithm based on the latter approach. Some relations among the search directions and the residuals, and how the search directions are related to the Krylov subspace are investigated. Numerical experiments are reported to verify the convergence properties.     

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.     

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ²⁺¹→S ² of the form where u is the polar angle on the sphere, is the ground state harmonic map, λ(t)=t ^-1-ν, and is a radiative error with local energy going to zero as t→0. The number can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis. Mathematics Subject Classification (1991) 35L05, 35Q75, 35P25     

On stable self‐similar blowup for equivariant wave maps

We consider corotational wave maps from (3 + 1) Minkowski space into the 3‐sphere. This is an energy supercritical model that is known to exhibit finite‐time blowup via self‐similar solutions. The ground state self‐similar solution f₀ is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f₀ is stable under the assumption that f₀ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f₀ rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f₀. © 2011 Wiley Periodicals, Inc.     

Multiplicity results for two kinds of equivariant systems

Two equivariant problems of the form $\epsilon \mathrm{\Delta }u=\nabla F\left(u\right)$ are considered, where $F$ is a real function which is invariant under the action of a group $G$, and, using Morse theory, for each problem an arbitrarily great number of orbits of solutions is founded, choosing $\epsilon$ suitably small.The first problem is a $O\left(2\right)$-equivariant system of two equations, which can be seen as a complex Ginzburg-Landau equation, while the second one is a system of $m$ equations which is equivariant for the action of a finite group of real orthogonal matrices $m\times m$.     

Harmonic extensions of symmetric maps

We examine the behavior of the Douady–Earle extension and we prove that a symmetric self-map of the circle can be harmonically extended to a conformal self-map of the hyperbolic disc.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

具有旋转对称根的多项式的牛顿映照

主要研究特殊多项式的牛顿映照的动力学性质.通过研究根的分布和重数,揭示了当多项式的根关于某点具有一定的旋转对称性,且对称根的重数都相同时,此类多项式的牛顿映照要么是双曲的,要么是次双曲的.另外多项式的牛顿映照的动力学性质为多项式的某些问题提供了新的思路.