# Andrew H. OsbaldestinUniversity of Portsmouth · Mathematics

Andrew H. Osbaldestin

PhD

## About

51

Publications

2,528

Reads

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390

Citations

Citations since 2016

Introduction

**Skills and Expertise**

Additional affiliations

February 2003 - present

October 1985 - January 2003

October 1983 - September 1985

**Queen Mary College, University of London**

Position

- PostDoc Position

Education

October 1980 - September 1983

## Publications

Publications (51)

We provide a renormalization analysis of correlations in a quasi-periodically forced two-level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function. For certain intensities of modulation, we give a complete understanding of the autocorrelation function. Furthermore, on...

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormaliz...

We provide evidence that the box-counting dimension of a structurally stable strange non-chaotic attractor (SNA) of pinched skew product type is equal to 2 by showing that it has non-negligible area. The argument presented is made more accurate in the study of a piecewise linear SNA. Furthermore we provide evidence that the fractal dimension of a c...

The full text of this article is available in the PDF provided.

We study autocorrelation functions in symmetric barrier billiards for golden mean trajectories with arbitrary barriers. Renormalization analysis reveals the presence of a chaotic invariant set and thus that, for a typical barrier, there are chaotic correlations. The chaotic renormalization set is the analogue of the so-called orchid that arises in...

We present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories. Depending on the nature of the barrier, this leads to either self-similar or chaotic behaviour. In the self-similar case we give an analysis of the half barrier and present a detailed calculation...

We construct a prototypical example of a spatially-open autonomous Hamiltonian system in which localised, but otherwise unbiased, ensembles of initial conditions break spatio-temporal symmetries in the subsequent ensemble dynamics, despite time reversal symmetry of the equations of motion. Together with transient chaos, this provides the mechanism...

We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in the Harper equation, including the renormalization strange set (known as the orchid) in the generalized Harper equation; and (ii) self-similarities in th...

We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: First, we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end, a homoclinic Melnikov a...

We explore the scattering of particles evolving in a two-degree-of-freedom Hamiltonian system, in which both degrees of freedom are open. Particles, initially having all kinetic energy, are sent into a so-called "interaction region," where there will be an exchange of energy with particles that are initially at rest. The open nature of both compone...

We propose a mechanism to rectify charge transport in the semiclassical
Holstein model. It is shown that localised initial conditions, associated with
a polaron solution, in conjunction with a nonreversion symmetric static
electron on-site potential constitute minimal prerequisites for the emergence
of a directed current in the underlying periodic...

A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained oscillators). The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another (S.P.~Ku...

We study the dynamics of particles evolving in a two-dimensional periodic, spatially-symmetric potential landscape. The system is subjected to weak external time-periodic forces rocking the potential in either direction which, inter alia, breaks integrability. In particular, chaotic layers arising around separatrices which connect unstable equilibr...

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in
a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit
of the chain. We study the instigation of directed transport in two types of system: (i) a microcanoni...

We propose a minimal model for the emergence of a directed flow in autonomous hamiltonian systems. It is shown that internal breaking of the spatiotemporal symmetries, via localized initial conditions, which are unbiased with respect to the transporting degree of freedom, and transient chaos conspire to form the physical mechanism for the occurrenc...

We study the autonomous Hamiltonian dynamics of non-interacting particles trapped initially in one well of a symmetric multiple-well washboard potential. The particles interact locally with an anharmonic oscillator acting as an energy deposit. For a range of interaction strengths, the particles gain sufficient energy during a chaotic transient to e...

We study the conservative and deterministic dynamics of two nonlinearly interacting particles evolving in a one-dimensional spatially periodic washboard potential. A weak tilt of the washboard potential is applied biasing one direction for particle transport. However, the tilt vanishes asymptotically in the direction of bias. Moreover, the total en...

A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytical maps (such as the Mandelbrot set and Julia sets). The system is composed of two alternately excited coupled oscillators. The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another [S.P. Kuznetsov, Example of a physical...

A model system of two coupled nonautonomous oscillators is proposed, in which the phenomena of complex analytic dynamics (Mandelbrot
and Julia sets, etc.) characteristic of complex logistic maps are realized. The idea underlying the model is based on the
mechanism of alternative excitation transfer from one subsystem to another and on the method (w...

The networks of globally coupled maps with a pacemaker have been introduced. We consider a generalization of the Kaneko model with a pacemaker represented by a single period-doubling element coupled unidirectionally with a set of other mutually coupled cells. We also investigate the dynamics of a system of two unidirectionally coupled elements, whi...

We consider the dynamics of small perturbations of stable two-frequency quasiperiodic orbits on an attracting torus in the quasiperiodically forced Henon map. Such dynamics consists in an exponential decay of the radial component and in a complex behaviour of the angle component. This behaviour may be two- or three-frequency quasiperiodicity, or it...

A transition from a smooth torus to a chaotic attractor in quasiperiodically forced dissipative systems may occur after a finite number of torus-doubling bifurcations. In this paper we investigate the underlying bifurcational mechanism, which is responsible for the termination of the torus-doubling cascades on the routes to chaos in invertible maps...

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 str...

We consider a generalized Harper equation at quadratic irrational flux, showing, in the strong coupling limit, the fluctuations of the exponentially decaying eigenfunctions are governed by the dynamics of a renormalization operator on a renormalization strange set. This work generalizes previous analyses which have considered only the golden mean c...

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 p...

We establish rigorous bounds for the unstable eigenvalue of the period-doubling renormalization operator for asymmetric unimodal maps. Herglotz-function techniques and cone invariance ideas are used. Our result generalizes an established result for conventional period doubling.

As shown recently [O.B. Isaeva et al., Phys. Rev. E 64, 055201 (2001)], the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems. We study scaling regularities associated with the effect of additive noise upon the period-tripling bifurcation cascade generalizing the renormalization grou...

This pre-print has been submitted, and accepted, to the journal, Physica D - Nonlinear Phenomena [© Elsevier]. The definitive version: CHAPMAN, J.R. and OSBALDESTIN, A.H., 2003.Self-similar correlations in a barrier billiard. Physica D - Nonlinear Phenomena, 180(1-2), pp. 71-91, is available at: http://www.sciencedirect.com/science/journal/01672789...

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 m...

A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations with coefficients expressed in terms of Feigenbaum's universal fixed-point function. The accuracy of the results depends...

In this paper, we develop a new approach to the construction of solutions of the Feigenbaum–Cvitanović equation whose existence was shown by H. Epstein. Our main tool is the analytic theory of continued fractions. To cite this article: A.V. Tsygvintsev et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 683–688.

We calculate all piecewise-constant periodic orbits (with values ±1) of the renormalisation recursion arising in the analysis of correlations of the orbit of a point on a strange nonchaotic attractor. Our results make rigorous and generalise previous numerical results.

This is a pre-print. The definitive version: MESTEL, B.D., OSBALDESTIN, A.H. and WINN, B., 2000. Golden mean renormalization for the Harper equation: The strong coupling fixed point. Journal of Mathematical Physics, 41(12), pp.8304-8330. We construct a renormalisation fixed point corresponding to the strong coupling limit of the golden mean Harper...

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation...

This is a pre-print. The definitive version: MESTEL, B.D. and OSBALDESTIN, A.H., 2000. Asymptotics of scaling parameters for period-doubling in unimodal maps with asymmetric critical points. Journal of Mathematical Physics, 41 (7), pp.4732-4746, is available at: http://jmp.aip.org/. The universal period-doubling scaling of a unimodal map with an as...

Using thermodynamical formalism, we set up the perturbative approach for the Hausdorff dimension and the dimension spectrum of the Julia set associated with polynomials close to monomials. We extend previous calculations to a more general case where the perturbation is not constant. 1. Introduction We consider the dynamics of the complex polynomial...

We carry out a renormalization analysis for unimodal maps possessing a degree-d critical point with differing left and right dth derivatives. More precisely, we prove, using Herglotz function techniques, the existence of a family of period-two points of the Feigenbaum renormalization operator. These universal functions (and their associated scaling...

The general Z4 and Z5 models on the simple quadratic lattice, and the pair triplet Ising model on the triangular lattice are employed to illustrate an extension of the finite size scaling technique which obtains the qualitative structure of the phase equilibrium surface in terms of multiple phase coexistence.

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.

It is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second-order transition points. The transformation when defined on finite or semi-infinite systems will not distinguish between order-disorde...

The authors numerically determine the f( alpha ) spectrum of scaling indices for critical KAM tori in the area-preserving standard map. Similarities and differences between the corresponding orbits of circle maps are discussed.

It is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible. The method is illustrated by an application to the three-, four- and five-state Potts models, and even in...

The author numerically determines the generalized dimensions and spectrum of singularities of a critical point orbit on the boundary of a Siegel disc. The local scaling exponent, alpha MN, and spectrum dependence on the degree, d, of the critical point are also considered. He finds that limd to infinity mod alpha MN mod d approximately=0.523 53 and...

We calculate rigorous bounds on the Hausdorff dimension of Siegel disc boundaries for maps that are attracted to the critical fixed point of the renormalization operator. This is done by expressing (a piece of) the universal invariant curve of the fixed-point maps as the limit set of an iterated function system. In particular, we prove (by computer...

We apply the methods of H. Epstein to prove the existence of a line of period-2 solutions of the Feigenbaum period-doubling renormalisation transformation. These solutions govern the universal behaviour of families of unimodal maps with “asymmetric critical points” of degree d, for which the d
th derivative has differing left and right limits.

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary
of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that
the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalizat...

We extend the renormalization analysis for period doubling in unimodal maps to the case of asymmetric critical points. Universal scaling phenomena are governed by period-two points of a renormalization operator.

We show that it is possible to use Taylor series approximations to numerically find scaling functions associated with the Julia sets of complex maps. The example we use is the mapping z→z2+c. The Taylor series are in powers of the parameter c about the trivial case c=0.

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 ar...

Iterated complex maps defined by a root of a biquadratic equation g(z', z) = 0 are studied. Numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the Riemann sphere. An analogy with Hamiltonian systems is discussed.

A phase equilibrium surface for the 2D (square lattice) antiferromagnetic Potts model in the presence of its ordering fields has been constructed using the phenomenological scaling transformation.