Robert S. MacKay

Transport in Hamiltonian systems

Abstract We develop a theory of transport in Hamiltonian systems in the context of iteration of area-preserving maps. Invariant closed curves present complete barriers to transport, but in regions without such curves there are still invariant Cantor sets named cantori, which appear to form partial barriers. The flux through the gaps of the cantori is given by Mathers differences in action. This gives useful bounds on transport between regions, and for one parameter families of maps it provides a universal scaling law when a curve has just broken. The bounds and scaling law both agree well with numerical experiment of Chirikov and help to explain an apparent disagreement with results of Greene. By dividing the irregular components of phase space into regions separated by the strongest partial barriers, and assuming that the motion is mixing within these regions, we present a global picture of transport, and indicate how it can be used, for example, to predict confinement times and to explain longtime tails in the decay of correlations.

Renormalisation in Area-Preserving Maps

Part 1 Introduction to area-preserving maps: conservative systems and maps periodic orbits invariant circles stochastic behaviour. Part 2 Introduction to renormalization: renormalization in physics renormalization in dynamical systems renormalization techniques. Part 3 Period doubling in area-preserving maps: period doubling sequencers renormalization the universal one parameter family appendices. Part 4 Renormalization for invariant circles: renormalization fixed point analysis simple fixed point critical fixed point the universal one parameter family discussion renormalization for maps on a circle.

A renormalization approach to invariant circles in area-preserving maps

Abstract Kadanoff and Shenker introduced a renormalisation approach to invariant circles in area-preserving maps. This paper makes more precise the connection between invariant circles and the renormalisation operator. Restricting attention to noble rotation numbers, the stability of a simple fixed point of the renormalisation is analysed, corresponding to a linear twist map. It is found to be essentially attracting, so that noble circles persist under perturbation, giving a new view on KAM theory. Shenker and Kadanoff found evidence for another fixed point, corresponding to a map with a non-smooth noble circle. Further evidence is given in this paper. It has essentially only one unstable direction, and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle. Finally, noble circles are shown to be locally most robust, in an important sense.

Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Abstract Many systems exhibit behaviour which can be described by three coupled oscillators. Provided the coupling is not too strong, such systems can be modelled by maps of the two-torus to itself. In this paper we describe and explain those aspects of the bifurcation diagram for two-parameter families of torus maps ƒ that involve change of mode-locking type. We introduce the concepts of partial and full mode-locking . For the coupled oscillators, these notions correspond to the presence of one or two rational relations between the frequencies, respectively. Numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations . In order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time-1 map of a flow on the torus. Then we analyse those condimension-1 and -2 bifurcations for flows on the torus which change the set of frequency ratios. We find a large variety of bifurcation diagrams, in particular many involving homotopically non-trivial saddle connections. Next we ask how the picture changes when the time-1 maps of a family of flows are perturbed to a general family of diffeomorphisms. We find toroidal chaos in the neighbourhood of all these bifurcations, meaning that there exist orbits which perform a pseudo-random sequence of rotations in different directions around the torus. The corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk. Finally, we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps, which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation.

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of 1δ = 18.721097200…, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = −4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.

Converse KAM: theory and practice

We unify, extend, reinterpret and apply criteria of Birkhoff [1], Herman [9], Mather [2, 3], Aubry et al. [4, 5], and Newman and Percival [6] for the nonexistence of invariant circles for area preserving twist maps. The criteria enable one to establish regions of phase space through which no rotational invariant circles pass. For families of maps the same can be done for regions of the combined space of phase points and parameters. The criteria can be implemented rigorously on a computer, and give a practical method of proving quite strong results. As an example, we present a computer program which proved that the “standard map” has no rotational invariant circles for any parameter value |k|≧63/64.

Resonances in area-preserving maps

Abstract A resonance for an area-preserving map is a region of phase space delineated by “partial separatrices”, curves formed from pieces of the stable and unstable manifold of hyperbolic periodic points. Each resonance has a central periodic orbit, which may be elliptic or hyperbolic with reflection. The partial separatrices have turnstiles like the partial barriers formed from cantori. In this paper we show that the areas of the resonances, as well as the turnstile areas, can be obtained from the actions of homoclinic orbits. Numerical results on the scaling of areas of resonances with period and parameter are given. Computations show that the resonances completely fill phase space when there are no invariant circles. Indeed, we prove that the collection of all hyperbolic cantori together with their partial barriers occupies zero area.

Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two-, and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers.

Transition to topological chaos for circle maps

Abstract Many biperiodic flows can be modelled by maps of a circle to itself. For such maps the transition from zero to positive topological entropy can be achieved in several ways. We describe all the possible routes for smooth circle maps, and discuss the relevance of our results to the transition to chaos for two-frequency systems.

Stability of Water Waves

We apply some general results for Hamiltonian systems, depending on the notion of signature of eigenvalues, to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability, up to non-degeneracy assumptions. A new superharmonic instability is predicted and verified.