Integrable Maps and Canonical Transformations

Perhaps the simplest form of dynamical system is a map of some space into itself, interpreted as a discrete-time evolution. Maps of the real line to itself are so simple that they can be iterated on a hand-held calculator. This apparent simplicity masks the fact that maps generally behave in a much more complex fashion than continuous time dynamical systems (differential equations). Whilst the logistic map has a rich structure of bifurcations, leading to chaos, its differential counterpart is a simple, separable ordinary differential equation.

However, some maps have extra structure, which can render them integrable. For instance, a map may possess "commuting maps" (the discrete equivalent of symmetries or commuting flows) or be symplectic, whilst possessing enough invariant functions ("first integrals") in involution. The latter includes area preserving maps of the plane with a single invariant, which means that any iteration is restricted to a "level curve" of the invariant. Integrable maps often possess a "Lax pair", which enables a complicated, nonlinear mapping to be re-written in terms of a simple matrix equation. Information about commuting maps or first integrals is then "locked inside" the Lax matrix.

Integrable maps can be studied without any reference to differential equations, but they can be related to these through the continuum limit or as Bäcklund transformations or nonlinear superposition laws for integrable partial differential equations. Furthermore, through a Poisson bracket, first integrals generate continuous symmetries. Alternatively, the map, as a discrete symmetry of these continuous flows, can be considered as a Bäcklund transformation.

Specific research in this area is as follows

1. For many differential equations a Bäcklund transformation may be constructed from canonical transformations. This enables us to establish connections between systems of integrable differential and difference equations. There are many starting points. We may restrict Darboux transformations to stationary flows [1] or use the Hamilton-Jacobi equation to construct a canonical transformation which preserves a given Hamiltonian function [2]. Examples include (a special case of) the McMillan map and Bäcklund transformations for some of the Painlevé equations. It is also possible to seek canonical transformations which preserve two or more compatible Poisson brackets or to look at nonlinear group actions.
2. In [3] we give a general construction of 2n-dimensional maps with n first integrals, which naturally generalise the QRT map. Whilst it is not yet known whether all these maps are integrable, they do possess many interesting, integrable sub-families. Other related results can be found in [4] and will be published in due course.

References

1. A.P. Fordy, A.B. Shabat, and A.P. Veselov, Factorisation and Poisson correspondences. Theor.Math.Phys., 105 , 225-45, 1995.
2. A.P. Fordy, Integrable symplectic maps. In P.A. Clarkson and F,W. Nijhoff, editors, Proceedings of SIDE II , 1998.
3. A.P. Fordy and P.G. Kassotakis, Multidimensional maps of QRT type. J.Phys.A, 39 , 10773-86, 2006.
4. P.G. Kassotakis, The construction of discrete dynamical systems, PhD Thesis, University of Leeds, 2006.

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Last Updated: 28th April, 2007.