Stability and uniqueness of periodic orbits produced during homoclinic bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].Supported by the National Natural Science Foundation of China.     

Stability,attractors, and bifurcations of the A2 symmetric flow

The A2 symmetric flow, initially introduced to study effects of symmetry in chaos synchronization, displays a variety of attractors and bifurcations much richer than initially though. These are studied in this article by means of two approaches. A linear stability analysis is used to determine fixed points, the nature of its stability, and where oscillatory solutions are expected. Nonlinear techniques such as bifurcation diagrams, Lyapunov exponents and phase space plots, are used to find and classify these oscillations and their bifurcations.     

Spherically symmetric internal layers for activator-inhibitor systems II. Stability and symmetry breaking bifurcations

A reaction-diffusion system of activator-inhibitor type is studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. We analyze the stability property of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions which are not spherically symmetric.     

Critical orbits of symmetric functionals

Let G be a compact Lie group and V a G-module, i.e. a finite-dimensional real vector space on which G acts orthogonally. We are interested in finding G-orbits of critical points of G-invariant C²-functionals f: SV→—, SV the unit sphere of V. Using a generalization of the Borsuk-Ulam theorem by Komiya [15] we give lower bounds for the number of critical orbits with a given orbit type. These results are applied to nonlinear eigenvalue problems which are symmetric with respect to an action of O(3) or a closed subgroup of O(3).     

Invariants,orbits and symmetric spaces

Lavoro svolto nell'ambito del G.N.S.A.G.A. del C.N.R. con contributo del M.P.I. fondi 40%.     

Exploiting orbits in symmetric ILP

This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.Mathematics Subject Classification (2000):90C10, 90C27, 90C57     

Degenerate resonances and branching of periodic orbits

In this paper we establish results on the existence of Lyapunov families of periodic orbits of reversible systems in around an equilibrium that presents a 0:1:1-resonance. The main proofs are based on a combined use of normal form theory, Lyapunov–Schmidt reduction and elements of symbolic computation. This work was supported by France–Brazil cooperation.     

Singular limit cycle bifurcations to closed orbits and invariant tori

This paper investigates singular limit cycle bifurcations for a singularly perturbed system. Based on a series of transformations (the modified curvilinear coordinate, blow-up, and near-identity transformation) and bifurcation theory of periodic orbits and invariant tori, the bifurcations of closed orbits and invariant tori near singular limit cycles are discussed.     

Stability of singular Hopf bifurcations

A stability formula is given for the singular Hopf bifurcation arising in singularly perturbed systems of the form , in this paper. The derivation of the formula is based on a reduction technique and on an existing stability formula for Hopf bifurcation.     

Computation and parametrization of periodic and connecting orbits

Email: g.moore{at}ma.ic.ac.uk or na.gmoore New algorithms for the computation of periodic and connectingorbits of autonomous differential equations are developed. Previousmethods have parametrized these curves by the independent timevariable, but our procedure is based on the canonical arclengthparametrization.     

On almost periodic orbits and minimal sets

In this short note we solve in the negative the three problems recently posed by Jie-Hua Mai and Wei-Hua Sun regarding the behaviour of almost periodic orbits and minimal sets of dynamical systems whose phase space is not regular.     

On almost periodic orbits and minimal sets

In this short note we solve in the negative the three problems recently posed by Jie-Hua Mai and Wei-Hua Sun regarding the behaviour of almost periodic orbits and minimal sets of dynamical systems whose phase space is not regular.     

Foliated vector fields without periodic orbits

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.