John A G Roberts

Integrable mappings and soliton equations II

Abstract Some simple solutions of discrete soliton equations are shown to satisfy 2D mappings. We show that these belong to a recently introduced 18-parameter family of integrable reversible mappings of the plane, thus lending weight to a previous conjecture. We also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics. Finally we discuss the notion of (generalized) reversibility.

Integrable mappings and soliton equations

Abstract We report an 18-parameter family of integrable reversible mappings of the plane. These mappings are shown to occur in soliton theory and in statistical mechanics. We conjecture that all autonomous reductions of differential-difference soliton equations are integrable mappings.

Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems

Abstract Dynamical systems with independent (continuous or discrete) time variable t and phase space variable x are called reversible if they are invariant under the combination { t →− t , x → G x } where G is some transformation of phase space which is an involution ( G ∘ G = Identity ). Reversible systems generalise classical mechanical systems possessing time-reversal symmetry and are found in ordinary differential equations, partial differential equations and diffeomorphisms (mappings) modelling many physical problems. This report is an introduction to some of the properties of reversible systems, with particular emphasis on reversible mappings of the plane which illustrate many of their basic features. Reversible dynamical systems are shown to be similar to Hamiltonian systems because they can possess KAM tori, yet they are different because they can also have attractors and repellers. We create and study examples of these hybrid dynamical systems and discuss the question of how to recognise whether a given dynamical system is reversible.

Trace maps as 3D reversible dynamical systems with an invariant

AbstractOne link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so-called trace maps. A subclass of them are mappings on a one-parameter family of 2D surfaces that foliate ℝ3 (and also ℂ3). They are derived from transfer matrix approaches to properties of 1D quasicrystals. In this article, we consider various dynamical properties of trace maps. We first discuss the Fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2D surfaces. We highlight a particular surface where the motion is integrable and semiconjugate to an Anosov system (i.e., the mapping acts as a pseudo-Anosov map). We identify properties of symmetry and reversibility (time-reversal symmetry) in the Fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits. We show that a conservative period-boubling sequence can be identified when moving through the one-parameter family of 2D surfaces. By using generator trace maps, in terms of which all trace maps obtained from invertible two-letter substitution rules can be expressed, we show that many features of the Fibonacci trace map hold in general. The role of the Fricke character

Escaping orbits in trace maps

\hat I(x,y,z) = x^2 + y^2 + z^2 - 2xyz - 1

Integrable mappings of the plane preserving biquadratic invariant curves II

, its symmetry group, and reversibility for the Nielsen trace maps are described algebraically. Finally, we outline possible higher-dimensional generalizations.

Conservative and dissipative behaviour in reversible dynamical systems

We provide a general framework to construct integrable mappings of the plane that preserve a one-parameter family B(x,y,K) of biquadratic invariant curves where parametrization by K is very general. These mappings are reversible by construction (i.e. they are the composition of two involutions) and can be shown to be measure preserving. They generalize integrable maps previously given by McMillan and Quispel, Roberts and Thompson. By considering a transformation of the case of the symmetric biquadratic to a canonical form, we provide a normal form for the symmetric integrable map acting on each invariant curve. We give a Lax pair for a large subclass of our symmetric integrable maps, including at least a 10-parameter subfamily of the 12-parameter symmetric Quispel-Roberts-Thompson maps.

Reversible mappings of the plane

We study the finitely-generated group A of invertible polynomial mappings from C3 to itself (or R3 to itself) which preserve the Fricke-Vogt invariant I(x, y, z) = x2 + y2 + z2 − 2xyz − 1. Using properties of suitably-chosen generators, we give a necessary condition and sufficient conditions for infinite order elements of A to have an unbounded orbit escaping to infinity in forward or backward time. Our main motivation for this study is that A includes the so-called trace maps derived from transfer matrix approaches to various physical processes displaying non-periodicity in space or time. As shown previously, characterising escaping orbits leads to various conclusions for the physical model and vice versa (e.g. electronic properties of ID quasicrystals). Our results generalize in a simple constructive way those previously proved for Fibonacci-type trace maps.