Allan K. Evans

The Effects of Continuously Varying the Fractional Differential Order of Chaotic Nonlinear Systems

Abstract We consider nonlinear third order differential equations which are known to exhibit chaotic behaviour, and amend their order using fractional calculus techniques. By doing this we demonstrate that by continuously increasing the order of differentiation for those systems from 2 to 3, a period doubling route to chaos ensues. This period doubling begins at a system specific order value between 2 and 3.

Landau theory revisited

Abstract The Landau free energy GL(T,Q) is an important concept in structural phase transitions because it is often the meeting point of experiment and the development of microscopic models. We review recent work on the real Q and T dependence of GL, the origin of non-classical non-critical exponents for T below Tc, the cross-over from soft mode to O/D behaviour, local bifurcation at T* and relation to ordering kinetics. Framework structures such as many silicates and the perovskite structure allow special geometrical Rigid Unit phonon Modes. The material is then likely to be near to the soft mode limit, resulting in classical behaviour of Q(T) over a very wide T range. Long range correlations result in a very small Ginzburg interval.

Crossover between displacive and order/disorder behaviour in the Phi 4 model

The Phi4 potential is widely employed for modelling systems that undergo displacive phase transition of the soft mode or order/disorder. A molecular dynamic simulation is used to establish where the crossover lies between the two regimes, considering both limits of long range and short range intersite interactions. It appears that with both ranges of interaction the crossover occurs when the on-site potential changes from a single to a double well. The behaviour of the soft mode frequency omega is also investigated, and in particular the ratio of the slopes d omega 2/dT between T>Tc and T<Tc. The authors find this ratio attains the value -2 given by standard renormalized phonon theory only the double limit of long range coupling and extreme soft mode behaviour. The order parameter varies as (Tc-T)1/2 over a wide temperature range, as observed in several materials, in this double limit only.

Resolution limits and noise reduction in digital holographic microscopy

In digital holographic microscopy, a hologram of an object is recorded by an electronic image sensors and a computer is used to reconstruct the original object numerically. A number ro different arrangements have been successfully used, for example by Haddad, Schnars, Takaki and Jacquot. There is an intermediate case between the Fourier-transform method of Haddad et al and the Fresnel arrangement used by Jacquot et al, which has some of the advantages of both methods. A point reference source in a lane some distance from object provides the spatial frequency reduction in the hologram plane, as for Fourier transform methods, without the strong central peak and with the twin image defocused in the object plane. This arrangement is tested, showing that it can produce a resolution significantly improved over the Fresnel case. We also consider the removal of the holographic twin image in the Fresnel in-line holographic arrangement. The high-contrast parts of the image are assumed to be part of the true image, and the twin image corresponding to these high-contrast elements is subtracted, leaving an improved estimate of the true image only. We present experimental results demonstrating this method for a number of different objects.

Conditional entropy and randomness in financial time series

This paper investigates the fundamental question of whether or not financial time series can be predicted. It does so using the conditional entropy measure commonly used in information theory and statistical physics. This approach is appropriate because it will reveal patterns of arbitrary complexity. That is, this method will not just reveal linear correlation, or any specific nonlinear correlation, it will reveal any patterns in the data. Interesting discoveries include the fact that there is a degree of consistency amongst data sets and the ability to clearly distinguish between stock market indices and exchange rate time series. The fact that the magnitude of price movements is more correlated than the direction is verified and quantified in the context of daily data. The main result is that above certain time scales financial time series are random, but below this threshold there are situations where knowing a portion of the past can bring a statistically significant amount of certainty about the future.

Canonical nonequilibrium ensembles and subdynamics

A method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by Coveney and Penrose. In this paper, we show that the method is equivalent to the subdynamics formalism, developed by Prigogine and others, when the dimension of the subdynamic kinetic subspace is finite. The comparison between the two approaches helps to clarify the nature of the various operators used in the Brussels formalism. We discuss further the relationship between these two approaches, with particular reference to a simple discrete-time dynamical system, based on the bakers transformation, which we call the bakers urn.

Why study financial time series

This article gives an introduction to quantitative finance: why financial time series are interesting and what we stand to gain by studying them. The first part motivates and describes the role that complex systems, self-organised criticality and universality might play in unravelling the mystery surrounding the stylised and non-trivial empirical regularities observed in markets. The second part of the paper then describes each of these empirical regularities in turn with some mathematical background and popular statistical models.

The long-time behavior of correlation functions in dynamical systems

A correlation function can be interpreted physically as the expected result of an experiment on a dynamical system. The system is prepared in an initial state described by the probability density function b(x), allowed to evolve for time t, and then the quantity a(x) is measured. The long-time behavior of correlation functions often depends upon the smoothness of the functions a and b, both in chaotic and non-chaotic dynamical systems. An argument is presented showing how the long-time behavior of correlation functions is related to the smoothness of the phase-space functions involved and the divergence of trajectories. Finally, the results of some numerical experiments supporting these conclusions are described.

The stability of microbubbles and sonoluminescence

Sonoluminescence (SL), the light emission of a micron‐sized bubble of gas trapped in water by an acoustic field, is associated with the collapse of a bubble nonlinearly oscillating under the sound field. Recent advances have made it possible to trap a single bubble at the pressure antinode of a standing wave, where it collapses and reexpands with the periodicity of the applied sound and emits light with every cycle. A critical feature of SL is the spherical symmetry of the bubble. If the collapse is violent, irregularities in the spherical shape of the bubble may develop towards the end of the collapse. As a result, the bubble may get destroyed shortly after the collapse. Due to the microscopic size of the bubble, molecular fluctuations constitute an additional force to which the gas–liquid interface is exposed. This additional force causes irregularities in the spherical shape of the bubble. The stability of a bubble is discussed by analyzing results obtained from hydrodynamic calculations of the bubble dynamics and the fluid dynamical processes outside the bubble taking molecular fluctuations into consideration. Results of these calculations reveal a surprising stability behavior and are in good agreement with observed behavior in single‐bubble SL experiments. a)ua@dmu.ac.uk

CLASSICAL AND THERMODYNAMIC LIMITS IN A SYSTEM OF INTERACTING QUANTUM SPINS

We study a model system composed of interacting quantum spins, with every spin coupled to every other, in the limit where the number of spins K and the angular momentum j of each spin are both large, aiming to explore the effect of large system size on the breakdown of the classical limit of quantum mechanics. We obtain the exact spectrum of the Hamiltonian, and hence the trace of the quantum-mechanical time-evolution operator. We examine the time dependence of , finding a simple approximation which is valid when j and K are large. At a time proportional to , this approximation breaks down, and the long-time behaviour is extremely complex. We use a renormalization scheme to investigate this complexity. The scheme is based upon a generalization of the Gauss continued-fraction map to the complex plane.