Franco Vivaldi

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.

Arithmetical properties of strongly chaotic motions

Abstract The orbits of the generalized Arnold-Sinai cat maps, or hyperbolic automorphisms of the two-dimensional torus, typify purely chaotic, Anosov motion. We transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers, classify all periodic orbits, and show how to determine their periods and initial conditions. The methods are based on ideal theory in quadratic fields, which is reviewed. It is shown that the structure of orbits rests upon some basic arithmetical notions, such as unique factorization into prime ideals.

Global stability of a class of discontinuous dual billiards

An infinite-parameter family of discontinuous area-preserving maps is studied, using geometrical methods. Necessary and sufficient conditions are determined for the existence of some bounding invariant sets, which guarantee global stability. It is shown that under some additional constraints, all orbits become periodic, most of them Lyapounov stable, and with a maximal period in any bounded domain of phase space. This yields a class of maps acting on a discrete phase space.

Geometry of p -adic Siegel discs

Abstract We survey recent advances in the study of regular motions over p-adic fields, show its varied connections with dynamics and number theory, and illustrate its significance to an important class of discrete dynamical systems. We also show that mappings supporting quasi-periodic motions can be naturally interpreted as flows with p-adic time.

Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off

We investigate the effects of round-off errors on the orbits of a linear symplectic map of the plane, with rational rotation number nu=p/q. Uniform discretization transforms this map into a permutation of the integer lattice Z(2). We study in detail the case q=5, exploiting the correspondence between Z and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higher-dimensional embedding, we establish the quasi-periodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a long-range clustering property similar to that found in repetitive tilings of the plane. (c) 1997 American Institute of Physics.

Quadratic rational rotations of the torus and dual lattice maps

We develop a general formalism for computer-assisted proofs concerning the orbit structure of certain nonergodic piecewise affine maps of the torus, whose eigenvalues are roots of unity. For a specific class of maps, we prove that if the trace is a quadratic irrational (the simplest nontrivial case, comprising eight maps), then the periodic orbits are organized into finitely many renormalizable families, with exponentially increasing period, plus a finite number of exceptional families. The proof is based on exact computations with algebraic numbers, where units play the role of scaling parameters. Exploiting a duality existing between these maps and lattice maps representing rounded-off planar rotations, we establish the global periodicity of the latter systems, for a set of orbits of full density.

Recursive tiling and geometry of piecewise rotations by π/7

We study two piecewise affine maps on convex polygons, locally conjugate to a rotation by a multiple of ?/7. We obtain a finite-order recursive tiling of the phase space by return map sub-domains of triangles and periodic heptagonal domains (cells), with scaling factors given by algebraic units. This tiling allows one to construct efficiently periodic orbits of arbitrary period, and to obtain a convergent sequence of coverings of the closure of the discontinuity set ?. For every map for which such finite-order recursive tiling exists, we derive sufficient conditions for the equality of Hausdorff and box-counting dimensions, and for the existence of a finite, non-zero Hausdorff measure of . We then verify that these conditions apply to our models; we obtain an irreducible transcendental equation for the Hausdorff dimension involving fundamental units, and establish the existence of infinitely many disjoint invariant components of the residual set . We calculate numerically the asymptotic power law growth of the number of cells as a function of maximum return time, as well as the number of cells of diameter larger than a specified . In the latter case, the exponent is shown to coincide with the Hausdorff dimension.

Round-off errors and p-adic numbers

We explore some connections between round-off errors in linear planar rotations and algebraic number theory. We discretize a map on a lattice in such a way as to retain invertibility, restricting the system parameter (the trace) to rational values with power-prime denominator pn . We show that this system can be embedded into a smooth expansive dynamical system over the p -adic integers, consisting of multiplication by a unit composed with a Bernoulli shift. In this representation, the original round-off system corresponds to restriction to a dense subset of the p -adic integers. These constructs are based on symbolic dynamics and on the representation of the discrete phase space as a ring of integers in a quadratic number field.

Some p-adic representations of the Smale horseshoe

Abstract We extend one example of Verstegen on the p-adic representation of the chaotic doubling map to the Smale horseshoe map. We show that the horseshoe is topologically conjugate to a linear saddle-type diffeomorphism, defined on the product of two copies of the p-adic integers. We also find that there exist topological conjugacies of the horseshoe with homeomorphisms of a single copy of the p-adic integers, provided one relaxes the differentiability condition.

Anomalous transport in a model of Hamiltonian round-off

We study the propagation of round-off error near the periodic orbits of a linear area-preserving map - a planar rotation by a rational angle - which is discretized on a lattice in such a way as to retain invertibility. We consider the round-off error probability distribution as a function of time t, and we show that for each t this is an algebraic number, which can be calculated exactly. We prove that its kth moment increases asymptotically as , where is the fractional dimension of a self-similar set related to periodic orbits of long-period, while G is a bounded function, periodic in the logarithm of t. This implies the diffusion coefficient displays bounded variations, while all higher order transport coefficients diverge, resulting in anomalous transport. This result contrasts with the case of irrational rotations, where the existence of a central limit theorem has been recently established (Vladimirov I 1996 Preprint Deakin University).