Symmetry and Perturbation Theory

The following sections are included:

• Preface

• Acknowledgements

• CONTENTS

In this paper we focus on the Jacobi–Mumford system and its generalizations.

A method to deal with Hopf bifurcation in problems with O(2) symmetry is introduced. Application of the method on Kuramoto-Sivashinsky equation is considered and it is shown that a multiple Hopf bifurcation may occur on a branch with dihedral group of symmetry. This bifurcation is associated with the two dimensional irreducible representation of group D_n.

Smooth hamiltonian vector fields with linear symmetries and anti-symmetries are considered. We prove that provided symmetry group is compact then a smooth conjugacy in Sternberg-Chen Theorem can be chosen canonical and symmetric.

Topological instability of action variables in multidimensional nearly integrable Hamiltonian systems is known as Arnold Diffusion. This phenomenon was first pointed out in 1964 by Arnold himself for a model Hamiltonian in his famous paper¹. For autonomous Hamiltonian systems with two degrees of freedom KAM theory generically implies topological stability of the action variables (i.e. the time-evolution of the action variables for the perturbed system stay close to their initial values for all times). On the contrary, for systems with more than two degrees of freedom, outside a wide range of initial conditions (the so-called "Kolmogorov set" provided by KAM theory), the action variables may undergo a drift of order one in a very long, but finite time called the "diffusion time". After thirty years from Arnold's seminal work attention to Arnold diffusion has been renewed by Chierchia and Gallavotti⁶, using KAM theory and geometrical methods, and by Bessi⁵, using variational tools…

Note from Publisher: This article contains the abstract and references only.

Many models for physical phenomena in oceanography, atmospheric dynamics, optical fibre transmission, nerve conduction, acoustical and gas dynamic flows are conservative translation-invariant evolution equations with a Hamiltonian structure. Solitary waves and fronts form an important class of solutions of such equations and the calculus of variations, critical point theory and symplectic structure have played a major role in the analysis of their stability and instability. For example, the characterisation of solitary waves as critical points of the Hamiltonian (energy) constrained to level sets of the momentum (or momentum and other constants of motion) leads to a powerful framework for proving nonlinear Lyapunov stability – when the second variation, evaluated at the constrained critical point, has a finite number of negative eigenvalues (e.g. BENJAMIN², BONA³, HOLM ET AL¹⁴, GRILLAKIS ET AL^{12, 13}, MADDOCKS & SACHS¹⁶ and references therein)…

We apply the Lie-group formalism to deduce symmetries of the generalized Boussinesq equation,

The following section are included:

• Introduction and Main Results

• Acknowledgments

• References

This is an abstract of a paper, which will be published somwhere else. Our goal is to prove some qualitative results related to a generalization of Langmuir's problem, i.e. that of the classical isosceles 3-body problem (two electrons of equal masses and one nucleus, all assumed to be point particles) whose motion is due to a Coulomb force…

Note from Publisher: This article contains the abstract and references only.

In previous work⁵ we consider the existence and stability of heteroclinic cycles arising from local bifurcation in dynamical systems with wreath product symmetry , where Z₂ acts by ±1 on R, and is a transitive subgroup of the permutation group S_N (thus has degree N). The group Γ acts absolutely irreducibly on R^N. We consider primary (codimension one) bifurcations from an equilibrium to heteroclinic cycles as real eigenvalues pass through zero. We relate the possibility of such cycles to the existence of non-gradient equivariant vector fields of cubic order. Using Hilbert series and the software package MAGMA ¹ we show that apart from the cyclic groups (already studied by other authors) only five groups of degree ≤ 7 are candidates for the existence of heteroclinic cycles. We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discuss edge cycles, and a generalisation of heteroclinic cycles which we call a heteroclinic web. We apply our methods to three examples. The work briefly reported here was published in: Dynamics and Stability of Systems 15, 353-385 (2000).

Given a resonant Poincaré-Dulac normal form, we associate to it an auxiliary linear system; solutions to the original system are obtained from solutions to the auxiliary one on a certain invariant submanifold, defined by resonance conditions. If the linearization of the original system satisfies the Poincaré condition, the auxiliary system is finite dimensional.

In this paper we make a full analysis of the symmetry reductions of the (2 + 1)-dimensional integrable Schwarz-Korteweg-de Vries equation by using the classical Lie method of infinitesimals. The reduction to systems of partial differential equations in (1+1) are obtained from the optimal system of subalgebras. These systems admit symmetries which lead to further reductions, i.e. to systems of ordinary differential equations. Further, we present a brief analysis for some types of particular solutions.

Coupled Map Lattices (CMLs) are extended dynamical systems with discrete time and space (a lattice), while the field is continuous. Broadly speaking, a local dynamical system is located at each point of the space and interacts, via some spatial coupling, with a set of 'neighbouring' sites. Commonly, the local dynamics has strong chaotic properties…

Note from Publisher: This article contains the abstract and references only.

In Quantum Field Theory models of electro-weak interactions with spontaneously broken gauge invariance, renormalizability limits to four the degree of the Higgs potential, whose minimum determines the vacuum in tree approximation. Through the discussion of some simple variants of the Standard Model with two Higgs doublets, we show that the technical limit imposed by renormalizability may prevent the physical realizability of some phases of the system, that would be otherwise allowed by the symmetry of the Lagrangian of the system. We show that the incorporation into an effective Lagrangian of suitable composite particle fields may resolve this discrepancy.

A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented. Using an orbit space approach to the problem, we find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly determine, specifying the group-subgroup relations between bordering phases.

The concept of parent phase in phase transition theory and the advantages of this approach are elucidated by consideration of phase transition sequence in PrAlO₃.

We study symmetries of a 2+1-dimensional Burgers equation with variable coefficient. We show that the equation admits an infinite-dimensional Lie algebra as the algebra of its symmetry group which does not have a Virasoro structure whose presence characterize integrability for PDEs in more than 1+1-dimensions. We give a classification of its low-dimensional subalgebras and obtain reduced ODEs. In contrast to an integrable PDE, its reductions to ODEs do not lead to Painlevé type equations. We pick out of them those equations which pass the Painlevé test and obtain their exact solutions.

The Camassa-Holm equation of shallow water theory is generalized to fluid flow over a two-dimensional sea bottom. As in the one-dimensional case the resulting equations are of Euler-Poincaré type. The dynamics of the one-dimensional Camassa-Holm equation embeds into the dynamics of the new equations.

C∞–Symmetries are used to obtain a step by step method to reduce the order of equations that admit the non-solvable symmetry algebra sl(2, ℝ). A classification of third order equations with symmetry algebra sl(2, ℝ) is given, as well as the corresponding first order reduced equations.

Gordon's theorem claims that given a Hamiltonian system all of whose solutions are periodic, the period of a solution depend only on the value of the Hamiltonian function on the trajectory of this solution. Generalizations are obtained for the case of invariant isotropic tori of arbitrary dimension k (rather than k = 1), which fiber either all the phase space or some submanifold of this. One supposes that the system has some collection of k first integrals in involution, such that the corresponding vector fields are tangent to these tori. Then the frequencies of quasiperiodic motion on such a torus depend only on the values of these first integrals on the torus. This is true also for the circular action functions, but sufficient conditions are essentially different in these two cases.

The aim of this note is to survey the recent literature on the new equivariant theory of moving frames developed by the author and Mark Fels^14,15. The classical Cartan theory^11,18, as well as its more rigorous later revival^17,22, has a fairly limited range of geometrical applications. In contrast, the new equivariant theory can be systematically applied to completely general transformation groups, including infinite-dimensional Lie pseudo-groups. The full range of new applications is surprisingly broad, including complete classification of differential invariants and their syzygies, general equivalence and symmetry problems based on differential invariant and joint invariant signatures, classical invariant theory and algebra, computer vision and object recognition, the calculus of variations, Poisson geometry and solitons, and symmetry-based numerical approximation theory…

We give a few examples of how, using the knowledge available on the geometry of Hamiltonian dynamics with symmetry, standard critical point theory can be adapted to this setup in order to obtain predictions on the existence of various dynamical elements and, moreover, it can be used to provide estimates on the number of these solutions. The proofs of these results, as well as additional information, can be found in ^29,27

We study perturbed dynamical systems defined through an m–dimensional vector field. In this paper we illustrate how the computation of different normal forms of these systems leads to the calculation of different invariant manifolds of them.

In this review we study the periodic behaviour of some nonlinear PDEs looked upon as infinite dimensional dynamical systems. We briefly illustrate the origins and the motivations of this kind of problems and some of the lines along with they have been tackled. We explain then our contribution, which consists in a suitable combination of bifurcation and averaging techniques, leading to very simplified proofs and to results for completely resonant systems. Application to nonlinear string equation, and to some higher dimensional equations are given.

The Fermi Pasta Ulam chain with periodic boundary conditions admits discrete and continuous symmetries. These symmetries allow one to formulate important restrictions on the Birkhoff normal form of this Hamiltonian system. We derive integrability properties and KAM statements. Hence the combination of symmetry and resonance in the periodic Fermi Pasta Ulam chain explains its quasiperiodic behaviour. This article contains a summary of the results obtained in references ¹¹ and ¹²

We will consider Lagrangian differential equations of dimensions two and higher whose symmetry group generators contain an arbitrary function of one independent variable. We will study the relationship between symmetries, boundary conditions and local conservation laws, and discuss comparison with the Second Noether Theorem. We will demonstrate the conclusions on the example of the equation of nonstationary transonic gas flow.

We show how general ideas of Louis Michel on the application of topology, group theory, and commutative algebra in the qualitative analysis of symmetric dynamical systems can be concretized for finite-dimensional Hamiltonian dynamical systems with symmetries, notably atoms and molecules. This talk is a contribution to the special session in the memory of Louis Michel.

This paper reviews higher order resonance in two degrees of freedom Hamiltonian systems. We consider a positive semi-definite Hamiltonian around the origin. Using normal form theory, we give an estimate of the size of the domain where interesting dynamics takes place, which is an improvement of the one previously known. Using a geometric numerical integration approach, we investigate this in the elastic pendulum to find additional evidence that our estimate is sharp. In an extreme case of higher order resonance, we show that phase interaction between the degrees of freedom occurs on a short time-scale, although there is no energy interchange.

For a compact group G a relative equilibrium of a G-invariant Hamiltonian is orbitally Liapounov stable (G-stable) if the Hessian of an augmented Hamiltonian at the corresponding critical point is definite when restricted to the symplectic normal space. This is no longer true in general for proper actions of noncompact groups, essentially because the orbit space of the coadjoint action of the group need not be Hausdorff. This paper gives a summary of our results on the stability of Hamiltonian relative equilibria. It presents a more general stability criterion that applies to relative equilibria with 'tame' angular velocities, and also gives a result that sharpens G-stability to A-stability, where A is a subset of G that depends only on the momentum of the relative equilibrium.

A new kind of topologically unavoidable branch crossing in band structure of solids is established. It is proven that there exists a great variety of 4-branch energy bands in crystalline solids of the orthorhombic system in which the crossings are necessitated by symmetry and topology. These crossings are different from conventional degeneracies that follow from space group symmetry alone.

The Scientific Committee of the "Symmetry and Perturbation Theory 2001" conference has decided to organize a special session dedicated to the memory of Louis Michel, who passed away on 30 December, 1999…

Note from Publisher: This article contains the abstract and references only.

The following sections are included:

• CONFERENCE PROGRAM

• LIST OF PARTICIPANTS

• LIST OF TUTORIAL PAPERS

• SYMMETRY AND PERTURBATION THEORY – SPT 98 INTERNATIONAL WORKSHOP – ROMA 16-22/12/1998

• SYMMETRY AND PERTURBATION THEORY WORKSHOP – TORINO 15-20/12/1996