Strange Sets Research Articles

PEOPLE

ASE: Attend, Socialize, EnjoyBob Kibble reflects on the enriching effects of the annual meeting Bob Kibble is a teacher trainer at the University of Edinburgh, Scotland.I remember my first ASE meeting in Reading. Perhaps in 1978 or thereabouts. I had been teaching for a few years and thought I'd check out this local convention of science teachers. It was indeed a revelation that so many people had so much to say about teaching science. There was talk about N and F levels and the 'I level grill'. Someone had ordered something called a BBC machine (later revealed to me as the latest in hi-tech teaching). I remember it well. But it was a lonely affair for a recent recruit. People seemed to know each other and there was much friendly exchanging. However, nobody knew me and I knew nobody else. The professional revelations were accompanied by a personal isolation. A strange set of memories indeed for a new recruit, unskilled and clumsy in the social arena. Bob practising for the ASE singalong session this year. This year I went to the ASE Centenary meeting in Guildford, my sixteenth ASE annual meeting. Things have changed since the early days. Thursday started with a formal Cathedral service in celebration of 100 years of the ASE. I sat next to a lady from Oxford and behind my good friend Dave from Croydon. Things snowballed from there. I went to a workshop on the water cycle and was brought face to face with my own misconceptions about the life story of a water molecule. Got a freebie coloured bracelet as well. Thanks Margaret. A chap from Bournemouth gave me loads of ideas about how best to set up a shared lesson observation scheme as well as how to run a professional development workshop. Thanks Stuart. At a third session I joined Brenda from Cambridge and we spent an enjoyable hour discovering ways to approach the teaching of light and in particular Ibn al Haytham's revelations courtesy of a chap from Kingston.That afternoon I was invited to present a talk to teachers in Iceland and on Friday to teachers in Ireland. Fred and Frances from Southgate College showed great team spirit in the world's most chaotic quiz, then sang in harmony with Dave from Barnsley and Yoji from Japan amongst others.Kirsten from across the Atlantic shared her research on children and museums on Friday and later I discovered how much I didn't know about global warming from a chemist and a set of unreadable graphs in a computer lab. What a pleasure to see Tom and Seamus from Ireland again. The day closed with a barn dance fling where I managed to boogie on down with some teachers from Turkey. A final 'strip the willow' with Pete and Allison finished me off.Saturday started with breakfast with Ken from Greenwich offering me some part-time work. The exhibitions called and amongst others I hugged Sue from Cardiff, Barry and Leila from London, Pauline from Scotland and an unknown but very friendly lady from Holland. I met a lady from Scotland who lived in Iceland but who taught in Norway and of course Catherine was everywhere. I had tea with Jean from Edinburgh, Janet from London, Gordon from Aberdeen. I laughed with Mick from Sheffield and the same friendly lady from Holland. I ate posh food with Susie from Ayr and applauded big names from America and Australia. I shared a lump in my throat for John and for Wynne and nodded and smiled at countless people whose names I had forgotten but our paths had crossed sometime in the past.There are more than 20 000 members in the ASE and countless thousands of science teachers across the globe. The world is indeed expansive but for three days in Guildford it seemed that the very best speakers and the very cheeriest folk had magically converged on one place. If ever there were confirmation that science and science teaching were alive and well it was there and it was then. If you were there you will know exactly what I mean. If you weren't there then don't miss it next January in Liverpool. www.ase.org.uk

For my child’s future

Joel Regudo left Manila for Carrickfergus to make a better life for his family back home. Here, Joel tells his story and opposite his manager Rosemary Strange sets out the nursing home's welcome policy

Characteristics of a piecewise smooth area-preserving map.

We are reporting a study carried out in a system concatenated by two area-preserving maps. The system can be viewed as a model of an electronic relaxation oscillator with over-voltage protection. We found that a border-collision bifurcation may interrupt a period-doubling bifurcation cascade, and that some special features, such as "quasicoexisting periodic orbits crossing border" as well as the transition between "quasitransience" and chaotic orbits, accompany the process. These features belong to the so-called "quasidissipative" properties. Here "quasitransience" denotes the behavior of iterations outside elliptic islands. They are "attracted" to the islands. As soon as it reaches the islands, the iteration follows the conservative regulations exactly. This induces a kind of escaping from strange sets. The scaling behavior of the escaping rate is obtained numerically.

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.

THE PERIODICITY OF CHAOTIC IMPACT OSCILLATORS IN HAUSDORFF PHASE SPACES

It is well known that non-periodic behavior is one of the most puzzling characteristics of chaotic oscillators. So far chaotic dynamical systems have been investigated in Euclidean spaces. In this paper, the concept of non-autonomous dynamical systems and that of Hausdorff phase spaces are proposed. The behavior of chaotic impact oscillators is investigated in Hausdorff phase spaces. It is discovered that, although the non-autonomous dynamical systems described by chaotic impact oscillators are non-periodic in Euclidean phase spaces, they are periodic in Hausdorff phase spaces. This shows that Euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena. In addition, the periodicity of chaotic dynamical systems in Hausdorff metric spaces induces a new class of strange invariant sets in Euclidean spaces. Such strange invariant sets may be an ideal symbol of chaotic dynamical systems.

Semi-uniform ergodic theorems and applications to forced systems

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some `non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

Box-counting dimension without boxes: computing D0 from average expansion rates.

We propose an efficient iterative scheme for calculating the box-counting (capacity) dimension of a chaotic attractor in terms of its average expansion rates. Similar to the Kaplan-Yorke conjecture for the information dimension, this scheme provides a connection between a geometric property of a strange set and its underlying dynamical properties. Our conjecture is demonstrated analytically with an exactly solvable two-dimensional hyperbolic map, and numerically with a more complicated higher-dimensional nonhyperbolic map.

Complex dynamics in classical control systems

An analysis of the complex dynamical behavior of second-order linear plants controlled with conventional controllers is presented. The control signal is passed through a classical nonlinearity before being applied to the plant. Existence of periodic and homoclinic orbits is discussed. Using the Melnikov/Smale and Genesio/Tesi methods some conditions about the existence of invariant strange sets are also established. It is shown that simple classical control schemes with typical nonlinearities can exhibit chaotic dynamics in a certain range of the controller parameters. Numerical and experimental results support the analysis presented.

Multifractality at Transition to Chaos Complex-Temperature Singularities

It is shown, that multifractal complex-temperature singularities can play a significant role in the critical strange sets multifractality. These singularities lead to a finite radius of convergence of the real high-temperature expansions and, therefore, to necessity to use a finite-temperature expansions (an analytic continuation). It is shown, using analytic results on multifractality of strange attractors of the baker map and results of numerical computations of the multifractal spectra on all critical points of phase transitions from period-η-tupling to chaos in 1D iterative system (Chinese Phys. Lett.3, 285 (1986) and J. Phys.A25, 589 (1992)) as well as results of a recent numerical simulation of a quantum system with multifractal spectrum (J. Phys.A28, 2717 (1995)), that the finite-temperature expansions give good approximation for the generalized dimensions Dq in a representative interval of q.

Slow manifold structure and the emergence of mixed-mode oscillations

A detailed study of the slow manifold of a model exhibiting mixed-mode oscillations is presented. A scenario for the emergence of mixed-mode states which does not involve phase locking on a 2-torus is constructed. We show that mixed-modes correspond to the periodic orbits embedded in the horseshoe-type strange set and demonstrate how transformations of this set determine the transitions of mixed-mode states into each other.

Indecomposable Continua and the Characterization of Strange Sets in Nonlinear Dynamics

We discuss a topological property which we believe provides a useful conceptual characterization of a variety of strange sets occurring in nonlinear dynamics (e.g., strange attractors, fractal basin boundaries, and stable and unstable manifolds of chaotic saddles). Sets with this topological property are known as indecomposable continua. As an example, we give detailed results for the case of an indecomposable continuum that arises from the entrainment of dye advected by a fluid flowing past a cylinder. We show for this case that the indecomposable continuum persists in the presence of small noise.

Fractal characteristics of critical and localized states in incommensurate quantum systems

Incommensurate quantum systems with two competing periodicities exhibit metallic (with Bloch-type extended wave functions), insulating (with exponentially localized wave functions) as well ascritical (with fractal wave functions) phases. An exact renormalization method, which takes into account the inherent incommensurability, is used to obtain the phase diagram of various quantum models for the two-dimensional electron gas and for quantum spin chains in a magnetic field. In this approach, the scaling properties of the fractal eigenstates are characterized by a fixed point or a strange invariant set of the renormalization flow. One of our novel results is the existence of self-similar fluctuations in the localized states once the exponentially decaying envelope is factorized out. In almost all cases under investigation here, the universality classes can be broadly classified as those of the nearest-neighbor square or triangular lattices.

Theoretical and Numerical Validation of the Stochastic Interrogation Experimental Method

Stochastic interrogation is an experimental method that uses transient trajectories starting at nu merous pseudo-random initial conditions to obtain detailed information about the flow of a dynamical system in phase space. From this flow information, various global dynamical phenomena can be studied, such as the transition to complex basin boundaries, chaotic transients, and strange non-attracting sets. The existence of these features in turn allows the occurrence of a homoclinic bifurcation to be inferred, even when all attractors in a system are nonchaotic. In this paper, the validity of inferences made using the stochastic in terrogation experimental method is checked with the aid of a numerical model, using theoretical predictions from Melnikov theory and direct computations of invariant manifolds.

Escaping from Strange Sets in Discontinuous Circle Map

The leakage of iterations from chaotic attractor near crisis due to dynamical interaction between discontinuity and noninvertibility has been studied. The scaling law of the mean life time is derived analytically. The numerical result shows very good agreement.

Self-similarity and localization.

The localized eigenstates of the Harper equation exhibit universal self-similar fluctuations once the exponentially decaying part of a wave function is factorized out. For a fixed quantum state, we show that the whole localized phase is characterized by a single strong coupling fixed point of the renormalization equations. This fixed point also describes the generalized Harper model with next nearest neighbor interaction below a certain threshold. Above the threshold, the fluctuations in the generalized Harper model are described by a strange invariant set of the renormalization equations.

The Falkner-Skan Equation I. The Creation of Strange Invariant Sets

We study bifurcations in the Falkner-Skan equation. we show that for positive integer values of the parameter there are bifurcations which create an infinity of periodic and other interesting trajectories. These bifurcations provide a new mechanism for the creation of strange invariant sets in ordinary differential equations, and are co-dimension one.

Decay of metastable states with discrete dynamics.

We consider the escape from invariant sets of one-dimensional piecewise linear maps which are additively disturbed by weak Gaussian white noise. The escape rates from point attractors and from strange invariant sets in the vicinity of the crisis at fully developed chaos are analytically determined and compared with results from numerical simulations. Both situations are combined resulting in a model with a point attractor which has a strange invariant set as basin boundary. Numerically a nonexponential decay of the attractor is found that can be described by a Markovian three-state model with transition rates known from the previous analysis.

Time-series analysis of transient chaos.

A time-series analysis method of transient chaos is worked out which can also be applied to signals of laboratory experiments. The process is based on the construction of a long artificial time series obtained by gluing pieces of many transiently chaotic signals together. This artificial signal represents a long-time motion in the vicinity of the nonattracting chaotic set. Thus all of the well-known numerical methods developed for analyzing permanent chaotic behavior are applicable in a more convenient way than using many short separated time-series pieces. The method is illustrated and its validity is checked by the H\'enon map. The nonattracting strange set is reconstructed in the presence of both a periodic and a chaotic attractor, and quantitative characteristics such as dimensions and Lyapunov exponents are determined by means of time-delay embedding methods.

Extension of Melnikov criterion for the differential equation with complex function

The present paper presents an extension of Melnikov's theory for the differential equation with complex function. The sufficient condition for the existence of a homoclinic orbit in the solutions of a perturbed equation is given. The method shown in the paper is used to derive a precursor criterion for chaos. Suitable conditions are defined for the parameters of equations for which the equation possesses a strange attractor set. The analytical results are compared with numerical ones, and a good agreement is found between them.

Dynamical integral transform on fractal sets and the computation of entropy

We introduce an integral transform of wavelet type, which we call Dynamical Integral Transform, and we show that it can be used to compute the second Renyi entropy for a large class of invariant measures. The method is then generalized to the whole spectrum of the Renyi entropies and establishes a correspondence between thermodynamic formalism and the Dynamical Integral Transform of expanding strange sets. Numerical examples are presented.