Giuseppe Gaeta

Normal forms, symmetry and linearization of dynamical systems

We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter-dependent family - via the Poincare normal form approach. We discuss this formally at first, and later pay attention to the convergence of the linearizing procedure. We also discuss some generalizations of our main result.

Reduction of Poincaré Normal Forms

The Poincaré algorithm for reducing a system of ODEs to normal form (near an equilibrium point) is based on considering near-identity changes of coordinates generated by homogeneous polynomial functions hk. These are chosen in such a way as to eliminate the nonresonant terms of a corresponding order. We show that careful consideration of the higher-order terms generated in the transformation, and use of the arbitrarity in the choice of hk, permit us to obtain a significant simplification of the normal form. Our results are illustrated by a relevant example.

Discrete symmetries of differential equations

The determination of the continuous symmetries of differential equations follows a well known algorithm, and is reduced to solution of a set of linear equations; this is based on considering infinitesimal generators of the symmetries, so that the method does not extend to discrete symmetries. In this paper, we present a method to determine discrete symmetries in a certain class by means of a linear system, although this is considerably more difficult to solve than the one connected with continuous symmetries. We also consider the inverse (and simpler) problem of determining the most general equation admitting a given discrete symmetry. In the last part, we consider a number of examples, dealing in particular with symmetries of relevance to physics.

Poincare normal forms and Lie point symmetries

We study Poincare normal forms of vector fields in the presence of symmetry under general-i.e. not necessarily linear-diffeomorphisms. We show that it is possible to reduce both the vector field and the symmetry diffeomorphism to normal form by means of an algorithmic procedure similar to the usual one for Poincare normal forms without symmetry; this joint normal form can be given a simple geometric characterization.

Hopf-type bifurcation in the presence of multiple critical eigenvalues

The authors show that, assuming a weak form of symmetry in the equations, one can have Hopf-type bifurcation of periodic solutions even in the presence of doubly degenerate critical imaginary eigenvalues.

Non-perturbative linearization of dynamical systems

We consider the classical problem of linearizing a vector field X around a fixed point. We adopt a non-perturbative point of view, based on the symmetry properties of linear vector fields.

Commuting-flow symmetries and common solutions to differential equations with common symmetries

We point out that in certain cases, all the differential equations (for given indipendent and dependent variables) possessing a given symmetry necessarily share a common solution. Under weaker conditions, all such differential equations have a solution - in general, different for different equations - characterized by a common symmetry. We characterize this situation and the common solution, or the common symmetry of solutions, and give concrete examples.

Counting Symmetry-Breaking Solutions to Symmetric Variational Problems

By combining Michels geometric theory of symmetry breaking and classical results from variational analysis, we obtain a lower bound on the number of critical points with given symmetryH ⊑G of a potential symmetric underG. The result is obtained by applying the Ljusternik-Schnirelman category in the group orbit space, and can be extended along the same lines to more general situations.