M. Dellnitz, M. Golubitsky and M. Nicol

Symmetry of attractors and the Karhunen-Loeve decomposition

In: Trends and Perspectives in Applied Mathematics. (L. Sirovich, ed.) Appl. Math. Sci. 100 Springer-Verlag, New York, 1994, 73-108.

• Abstract

Recent fluid dynamics experiments have shown that the symmetry of attractors can manifest itself through the existence of spatially regular patterns in the time-average of an appropriate observable such as the intensity of transmitted light in the Faraday experiment (Gollub and co-workers). Dellnitz, Golubitsky and Melbourne also observed that symmetries of attractors of PDEs in phase space should manifest themselves as symmetry invariants of the time-average of the solution. This possibility was verified in certain numerically computed solutions of the Brusselator and the complex Ginzburg-Landau equations --- both reaction-diffusion systems defined on the unit interval.

In this paper we discuss how the symmetry of attractors can be detected numerically in solutions of symmetric PDEs and how symmetry considerations affect the appropriateness of a popular method for computing asymptotic dynamics in PDEs --- the Karhunen-Loeve decomposition.

In Barany, Dellnitz and Golubitsky a method for computing numerically the symmetry of attractors, based on the notion of detectives, was developed. The idea behind detectives is to transfer the question of determining the symmetry of a set (the attractor) in phase space to the problem of determining the symmetry of a point in some auxiliary space determined by the symmetry of the dynamical system. This technique was then proved to give the correct symmetry for open sets --- at least generically --- and was also implemented to show that it could work in practice. In this paper we prove a variant of the detective theorem under the assumption of a Sinai-Bowen-Ruelle (SBR) measure on the attractor in phase space.

As mentioned previously there is a popular method for computing the long term dynamics of a system of PDEs and of constructing sets of model ODEs for these dynamics and, indeed, for any time series. The Karhunen-Loeve decomposition proceeds by finding an orthogonal set of eigenfunctions that is well-suited to the data --- eigenfunctions that capture in decreasing order most of the `kinetic energy' of the system. The data is then expanded in terms of these eigenfunctions at each moment in time and the time variation of the coefficients describes the dynamics. To obtain a system of ODEs the eigenfunction expansion is truncated at some finite order --- thus obtaining a sophisticated Galerkin type approximation to the dynamics.

The importance of symmetry for the Karhunen-Loeve decomposition was emphasized in the work of Sirovich. and Berkooz. We have investigated how well the symmetry of the attractor used to generate the Karhunen-Loeve decomposition is reflected in the end result. We will show that the Karhunen-Loeve operator is equivariant with respect to the symmetry group of the underlying attractor. We have also found that the symmetry property of the Karhunen-Loeve decomposition does not always exactly reflect the symmetry properties of the underlying attractor. In fact, in some important cases, there is more symmetry introduced into the reduced system of ODEs than is present in the data. For instance, we will show that an SO(2) symmetric attractor of a scalar PDE on the line with periodic boundary conditions automatically leads to a reduced system which has O(2) symmetry.

We then suggest an extension of the Karhunen-Loeve decomposition that is guaranteed to have the correct symmetry properties and show that this extension can be viewed as a construction of a detective for this case. Our results suggest that the method for constructing an appropriate reduced system via a Karhunen-Loeve decomposition should always be combined with the computation of the symmetry type of the underlying attractor using detectives.

We now outline the structure of this paper. In Section 2 we discuss the results on detectives given in Barany, Dellnitz and Golubitsky and introduce SBR measures. In Section 3 we prove that ergodic sums also provide a method for constructing detectives. We then interpret these results for systems of PDEs in Section 4.

The remainder of the paper discusses symmetry aspects of the Karhunen-Loeve decomposition. In Section 5 we describe the standard Karhunen-Loeve decomposition and in Section 6 we show how the symmetries of an attractor for a PDE system are inherited by the Karhunen-Loeve decomposition. We note that the Karhunen-Loeve decomposition has at least the symmetries of the PDE attractor; as noted previously it may have more. In Section 7 we show how to modify this method so that it will produce the correct symmetries. This technique is based on the theory of detectives of Section 8. In the last two sections we discuss the symmetry of the reduced (Galerkin type) system of ODEs produced by the Karhunen-Loeve decomposition (Section 9) and present an example --- the Kuramoto-Sivashinsky equation (Section 10).