Near-critical behavior of the Zhong-Zhang model.

The Zhong-Zhang (ZZ) model is a one-degree-of-freedom dynamical system describing the motion of an insulating plate of length d floating on the upper surface of a convecting fluid, with locking at the boundaries. In the absence of noise, the system away from the boundaries is described by linear differential equations with a delay time τ. The d,τ plane consists of two domains separated by a critical curve. For asymptotically long times, subcritical orbits approach a nontrivial periodic attractor, while the supercritical ones tend to a stationary state at the origin. We investigate near-critical behavior using a modified fourth-order Runge-Kutta integration scheme. We then construct a piecewise analytic decomposition of the periodic attractor, which makes possible a far higher level of accuracy. Our results provide solid evidence for an asymptotic power-law approach to criticality of several observables. The power laws are fed back to determine the piecewise-analytic structure deep into the near-critical regime. In an Appendix, we explore the effect of introducing noise using modified order-3/2 Kloeden-Platen-Schurz stochastic integration, following several observable quantities through the near-critical parameter domain.

Renormalizable two-parameter piecewise isometries.

We exhibit two distinct renormalization scenarios for two-parameter piecewise isometries, based on 2π/5 rotations of a rhombus and parameter-dependent translations. Both scenarios rely on the recently established renormalizability of a one-parameter triangle map, which takes place if and only if the parameter belongs to the algebraic number field K=Q(5) associated with the rotation matrix. With two parameters, features emerge which have no counterpart in the single-parameter model. In the first scenario, we show that renormalizability is no longer rigid: whereas one of the two parameters is restricted to K, the second parameter can vary continuously over a real interval without destroying self-similarity. The mechanism involves neighbouring atoms which recombine after traversing distinct return paths. We show that this phenomenon also occurs in the simpler context of Rauzy-Veech renormalization of interval exchange transformations, here regarded as parametric piecewise isometries on a real interval. We explore this analogy in some detail. In the second scenario, which involves two-parameter deformations of a three-parameter rhombus map, we exhibit a weak form of rigidity. The phase space splits into several (non-convex) invariant components, on each of which the renormalization still has a free parameter. However, the foliations of the different components are transversal in parameter space; as a result, simultaneous self-similarity of the component maps requires that both of the original parameters belong to the field K.

Flights in a pseudo-chaotic system.

We consider the problem of transport in a one-parameter family of piecewise rotations of the torus, for rotation number approaching 1∕4. This is a zero-entropy system which in this limit exhibits a divided phase space, with island chains immersed in a "pseudo-chaotic" region. We identify a novel mechanism for long-range transport, namely the adiabatic destruction of accelerator-mode islands. This process originates from the approximate translational invariance of the phase space and leads to long flights of linear motion, for a significant measure of initial conditions. We show that the asymptotic probability distribution of the flight lengths is determined by the geometric properties of a partition of the accelerator-mode island associated with the flight. We establish the existence of flights travelling distances of order O(1) in phase space. We provide evidence for the existence of a scattering process that connects flights travelling in opposite directions.

Embedding dynamics for round-off errors near a periodic orbit.

We study the propagation of round-off errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the two-dimensional discrete phase space (a lattice) in a higher-dimensional torus, where points sharing the same round-off error are uniformly distributed within finitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coefficients. This representation affords efficient algorithms for classifying and computing the orbits and their exact densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [Lowenstein et al., Chaos 7, 49-66 (1997)] disappears, and that the growth of the number of orbits with the period is algebraic.(c) 2000 American Institute of Physics.

Fixed-point densities for a quasiperiodic kicked-oscillator map.

Suppose that a one-dimensional harmonic oscillator is subjected to instantaneous kicks q times per natural period, with the kick amplitude varying sinusoidally with position. Viewed stroboscopically in phase space, the motion has an infinitely extended periodic or quasiperiodic array of fixed points, as well as an infinite web of chaotic orbits. In the present work (restricted to the quasiperiodic case q=5) the fixed points are classified according to their local linear behavior, which depends essentially on a single variable, the residue R. With the aid of a five-dimensional embedding, a function rho(R) is calculated which for infinitesimal DeltaR gives the average density of fixed points in the plane with residue in the range (R,R+DeltaR). The location and strength of the singularities and discontinuities of rho(R) are extracted from relatively simple transcendental equations, and this makes possible efficient numerical determination of rho(R). An exact equality for the densities of positive-R and negative-R fixed points is proved using decagonal symmetry and the integral representation of rho(R). For parameter values below the period-doubling threshold, there are no unstable fixed points with R greater, similar 0, and so we have equality of the densities of stable centers and unstable saddles. (c) 1995 American Institute of Physics.

Distribution of fixed-point energies of a quasiperiodic Hamiltonian flow.

Energy distributions rho(+/-)(E) for the elliptic and hyperbolic fixed points of the Hamiltonian H(x,y)= summation operator (k=0) (4) cos [x cos(2pik/5)+y sin(2pik/5)] are calculated as integrals over a one-dimensional manifold M(E) in five-dimensional space. Singular points of M(E) produce three logarithmic singularities of rho(+/-)(E), and vanishing of connected components of M(E) gives rise to three discontinuities. The strengths of the singularities and discontinuities of rho(+/-)(E) are determined analytically, and the distributions are evaluated numerically for representative points in the nonsingular intervals. The calculation provides an explicit realization of general theorems concerning the critical points of infinitely smooth functions defined on an n-dimensional torus and restricted to a k-dimensional linear subset. Formally the calculation resembles the determination of the density of states of a dynamical system with one degree of freedom on a 2-torus, but with important differences due to topology and symmetry.

Interpolating Hamiltonians for a stochastic-web map with quasicrystalline symmetry.

A systematic Hamiltonian approximation scheme is developed for a stochastic-web map with fivefold quasicrystalline symmetry. Interpolating Hamiltonians are calculated up to tenth order in the control parameter a. The higher order Hamiltonians are used to provide bounds for closed invariant curves of the map, and to investigate the structural evolution of map's phase portrait for a

Parameter dependence of stochastic layers in a quasicrystalline web.

Stochastic web maps with approximate quasicrystalline symmetry possess an infinite number of inequivalent fixed points embedded in stochastic layers of varying thickness. In this investigation exploratory steps are taken toward a systematic numerical determination of the widths of the stochastic layers as a function of the web map's control parameter. The study concentrates on a particular stochastic layer in the approximately fivefold symmetric web. Computer graphics and a simple stretching-and-folding criterion provide a coarse view, which is supplemented at finer scales by Greene's residue method. The exact reflection symmetries of invariant sets, as well as a five-dimensional representation of the map, are exploited to improve numerical precision. As the control parameter varies, one finds not only variations expected from island chain structures, but also larger-scale oscillations whose origin is not understood.