Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation and periodic orbits for dissipative twist maps

I study two quite different problems in this thesis. Both have been written up as papers, with their own detailed introductions. The thesis essentially consists of these two papers, although the first paper has been extended for this purpose to include extra explanatory material. In this summary I give a less formal introduction to the two papers. The aim is to give the reader some idea as to how these papers came into being. 1. Symbolic Dynamics for the Renormalization of a Quasiperiodic Schrödinger Equation. The subject of dynamical systems first captured my imagination on a reading of Feigenbaum’s renormalization theory of universality in period doubling. David suggested that I work on a renormalization theory for a one-dimensional Schrödinger equation with quasiperiodic potential, there being very little known about the problem. These equations are very interesting for applications, e.g. stability analysis of Saturn’s rings, theory of an electron in a two- dimensional crystal with superimposed magnetic field, and electronic structure of quasicrystals, to mention but a few. David pointed out a paper by Kadanoff concerning a particularly simple example of a quasiperiodic Schrödinger equation, for which the renormalization map is two-dimensional. He suggested that if the map could be shown to possess a Smale horseshoe, then the spectrum of the Schrödinger equation must contain a Cantor set. It was relatively easy to deduce that this picture was correct by performing some computer experiments on the renormalization map. In fact I found more to be true: the dynamics of the map could be completely described using six symbols. Strings of these six symbols can then be used to label the spectrum. The proof of these facts is geometrical in spirit. It is a lengthy exercise in applying standard techniques developed for proving the existence of invariant Cantor sets in non-linear maps. The fact that precisely six symbols are required for the description of the renormalization map seems to be the consequence of the occurrence of this symbolic dynamics in an "exactly solvable" map, related to the problem, which I describe. An interesting question then arises: what scaling properties of the spectra of the optimally approximating periodic equations can be deduced from the global dynamics of the renormalization map? It is, of course, well known that the existence of a fixed point in a renormalization map leads to a scaling law, with exponent governed by the expanding eigenvalue of the linearized map at the fixed point A remark by Newhouse, that the theory topological pressure would be relevant to the problem, was very helpful at this point I deduced the existence of a "global" scaling exponent, describing the total measure of the bands of the optimally approximating periodic systems. The exponent is obtained by taking a certain average of eigenvalues at all the periodic points of the renormalization map. Using the multiplicative ergodic theorem, I deduced the existence of an "ergodic" scaling exponent, which measures how the length of a "typical" band in the spectrum of a periodic system decreases as the period increases. Finally, I applied a theorem of McCluskey and Manning to deduce bounds on the Hausdorff dimension of the spectrum of the quasiperiodic equation in terms of the two exponents mentioned above. 2. Periodic Orbits for Dissipative Twist Maps. Periodic orbits can be proved to exist in area preserving maps of a cylinder by an elegant variational approach. The orbits are obtained as minima of a real valued function of many variables (the number of variables being equal to the period), subjected to periodic boundary conditions. Given that periodic orbits exist, from results of Hall and Katok one can deduce the existence of quasiperiodic orbits (with irrational rotation number), using only the twist hypothesis. David suggested that I look for periodic orbits in dissipative twist maps, via a variational approach devised by him. However the function that it was suggested I minimize was highly inhomogeneous in its variables, and it was inappropriate to apply periodic boundary conditions. Nevertheless, I explored the minimization problem on a computer, and found that results could be obtained using rigid boundary conditions, fixed by a parameter which is later varied. This lead naturally to a more powerful topological approach for deducing the existence of periodic orbits, which exploits the geometry of the twist and the topology of the cylinder in a particularly simple way. I thus obtained a theorem on the existence of periodic (and hence quasi- periodic) orbits in one parameter families of dissipative twist maps. Using this topological approach, I also obtained results on the allowed periodic motions which can occur on an attractor of an individual dissipative twist map. The result relies heavily on a pioneering paper of Birkhoffs. A key step is to introduce the concept of an "attractor with the intersection property", which is a generalization of a strange attractor. After I had written up this result, I found that P.le Calvez had obtained a closely related result a few months previously, using a similar topological criterion for the existence of periodic orbits. The paper presented here was rewritten to take this into account

Unheard voices: parents' and adolescents' experiences of multisystemic therapy for young offenders.

Government guidelines for mental health interventions emphasise the importance of taking young people's views into account. This review examines what is known from the adolescent's perspective in research investigating the outcome of psychological therapies. The literature in three mental health domains that are particularly relevant to adolescence is focussed on: anorexia nervosa, depression and antisocial behaviour. Both quantitative and qualitative studies are examined and what has been asked of adolescents is explored. This rev iew highlights what can be learnt from eliciting adolescents' views and considers how these v iews can better inform treatment

The non-linear response of the magnetosphere: 30 October 1978

Previous efforts to find evidence of deterministic nonlinear dynamics in the global geomagnetic system have treated the geomagnetic system as autonomous. However, the geomagnetic system is strongly driven by the stochastic solar wind. We consider the response of the magnetosphere, as given by the AE index, for one day when the IMF had a nearly constant southward value. Using both a series of non-linear statistics and non-linear prediction of the response to the input signal $v B_s$, we find that there is some evidence for deterministic non-linear response of the Earth's magnetosphere on that day.Comment: 4 pages, Postscript file compressed and uuencoded, made with uufiles scrip

Constrained-Realization Monte-Carlo Method for Hypothesis Testing

We compare two theoretically distinct approaches to generating artificial (or ``surrogate'') data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single ``best'' model to the original data, and then to generate surrogate data sets that are ``typical realizations'' of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is ``pivotal'' if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic.Comment: 19 pages, single spaced, all in one postscript file, figs included. Uncompressed .ps file is 425kB (sorry, it's over the 300kB recommendation). Also available on the WWW at http://nis-www.lanl.gov/~jt/Papers/ To appear in Physica

Nonlinear analysis of bivariate data with cross recurrence plots

We use the extension of the method of recurrence plots to cross recurrence plots (CRP) which enables a nonlinear analysis of bivariate data. To quantify CRPs, we develop further three measures of complexity mainly basing on diagonal structures in CRPs. The CRP analysis of prototypical model systems with nonlinear interactions demonstrates that this technique enables to find these nonlinear interrelations from bivariate time series, whereas linear correlation tests do not. Applying the CRP analysis to climatological data, we find a complex relationship between rainfall and El Nino data

Testing for Chaos in Deterministic Systems with Noise

Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the ``tangent space'' and ``direct method'' for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

Development of an acceptable and feasible self-management group for children, young people and families living with Type 1 diabetes.

AIMS: This study developed an acceptable and feasible self-management intervention that addresses the self-identified needs of children and young people with Type 1 diabetes and their parents. METHODS: Phase 1 reviewed previous interventions and interviewed the clinical team, young people and families. Phase 2 ran three age-matched focus groups with 11 families of children aged 8-16 years. Feedback was used to modify the workshop. Phase 3 evaluated feasibility of delivery, as well as the effects on metabolic control, quality of life and fear of hypoglycaemia, measured at baseline and 1-3 months post intervention. RESULTS: Eighty-nine families were invited to take part. Twenty-two (25%) participated in seven pilot groups (median age of young people 10 years, 36% girls). The intervention comprised a developmentally appropriate workshop for young people and parents addressing: (1) blood glucose control, (2) the potential impact of long-term high HbA1c , (3) the effects of 'hypos' and 'hypers', (4) self-management techniques and (5) talking confidently to people about diabetes. Participants were enthusiastic and positive about the workshop and would recommend it to others. Young people liked sharing ideas and meeting others with diabetes, while parents enjoyed listening to their children talk about their diabetes knowledge. CONCLUSIONS: Families living with Type 1 diabetes participated in developing a self-management group intervention. Although we demonstrated acceptability and feasibility, the pilot study results do not support the development of a randomized control trial to evaluate the effectiveness in improving HbA1c

Recurrence plot statistics and the effect of embedding

Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably determinism and entropy of line length distribution, to the correlation sum as a function of embedding dimension. These expressions are obtained by deriving the transformation which generates an embedded recurrence plot from an unembedded plot. A single unembedded recurrence plot thus provides the statistics of all possible embedded recurrence plots. If the correlation sum scales exponentially with embedding dimension, we show that these statistics are determined entirely by the exponent of the exponential. This explains the results of Iwanski and Bradley (Chaos 8 [1998] 861-871) who found that certain recurrence plot statistics are apparently invariant to embedding dimension for certain low-dimensional systems. We also examine the relationship between the mutual information content of two timeseries and the common recurrent structure seen in their recurrence plots. This allows time-localized contributions to mutual information to be visualized. This technique is demonstrated using geomagnetic index data; we show that the AU and AL geomagnetic indices share half their information, and find the timescale on which mutual features appear

Analyzing Multiple Nonlinear Time Series with Extended Granger Causality

Identifying causal relations among simultaneously acquired signals is an important problem in multivariate time series analysis. For linear stochastic systems Granger proposed a simple procedure called the Granger causality to detect such relations. In this work we consider nonlinear extensions of Granger's idea and refer to the result as Extended Granger Causality. A simple approach implementing the Extended Granger Causality is presented and applied to multiple chaotic time series and other types of nonlinear signals. In addition, for situations with three or more time series we propose a conditional Extended Granger Causality measure that enables us to determine whether the causal relation between two signals is direct or mediated by another process.Comment: 16 pages, 6 figure

Granger Causality and Cross Recurrence Plots in Rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarised intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarised intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarised intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.Comment: 10 pages, 7 figure