Andrew Osbaldestin

On linearization of the Ermakov system

Abstract It is shown that the nonlinear Ermakov system may be reduced to consideration of a pair of linear equations. Geometric aspects of the procedure along with analytic results pertaining to its inversion are noted. Graphical results are presented for a particular Ermakov system that arises in two-layer long wave theory.

Golden mean renormalization for the Harper equation: The strong coupling fixed point

We construct a renormalization fixed point corresponding to the strong coupling limit of the golden mean Harper equation. We give an analytic expression for this fixed point, establish its existence and uniqueness, and verify properties previously seen only in numerical calculations. The spectrum of the linearization of the renormalization operator at this fixed point is also explicitly determined. This strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems.

Renormalisation in implicit complex maps

Abstract We study the dynamics of a family of implicit complex maps restricted to an invariant circle. The renormalisation analysis for golden mean rotation number is given. We find a one-parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renormalisation transformation. These period-2 points are pairs of fractional linear maps.

Self-similar correlations in a barrier billiard

We give a renormalization analysis of the self-similarity of autocorrelation functions in symmetric barrier billiards for golden mean trajectories. For the special case of a half-barrier we present a rigorous calculation of the asymptotic height of the main peaks in the autocorrelation function. Fundamental to this work is a detailed analysis of a functional recurrence equation which has previously been used in the analysis of fluctuations in the Harper equation and of correlations in strange non-chaotic attractors and in quantum two-level systems.

Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

Continued fractions and solutions of the Feigenbaum–Cvitanović equation

In this paper, we develop a new approach to the construction of solutions of the Feigenbaum-Cvitanovic equation whose existence was shown by H.Epstein.Our main tool is the analytic theory of continued fractions.

Golden mean renormalization for a generalized Harper equation: The Ketoja–Satija orchid

We provide a rigorous analysis of the fluctuations of localized eigenstates in a generalized Harper equation with golden mean flux and with next-nearest-neighbor interactions. For next-nearest-neighbor interaction above a critical threshold, these self-similar fluctuations are characterized by orbits of a renormalization operator on a universal strange attractor, whose projection was dubbed the “orchid” by Ketoja and Satija [Phys. Rev. Lett. 75, 2762 (1995)]. We show that the attractor is given essentially by an embedding of a subshift of finite type, and give a description of its periodic orbits.

1/s-expansion for generalized dimensions in a hierarchical s-state Potts model

We consider the s-state Potts model on the diamond hierarchical lattice for large s. We show that the generalized dimensions Dq of the density of zeros supported by the associated Julia set are given by Dq = 1 - (q - 1)|s|-2/3/4 log 2+O(s-1). The information dimension D1 equals 1 to all orders.

Asymptotics of scaling parameters for period-doubling in unimodal maps with asymmetric critical points

The universal period-doubling scaling of a unimodal map with an asymmetric critical point is governed by a period-2 point of a renormalization operator. The period-2 point is parametrized by the degree of the critical point and the asymmetry modulus. In this paper we study the asymptotics of period-2 points and their associated scaling parameters in the singular limit of degree tending to 1.

Renormalization of correlations in a quasiperiodically forced two-level system: quadratic irrationals

Generalizing from the case of golden mean frequency to a wider class of quadratic irrationals, we extend our renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. We give a description of all piecewise-constant periodic orbits of an additive functional recurrence generalizing that present in the golden mean case. We establish a criterion for periodic orbits to be globally bounded, and also calculate the asymptotic height of the main peaks in the correlation function.