Annibal Figueiredo

SADE) A Maple package for the Symmetry Analysis of Differential Equations

Abstract We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie–Backlund and potential symmetries, invariant solutions, first-integrals, Nother theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODEs (previously implemented by the authors in the package QPSI) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given. Program summary Program title: SADE Catalogue identifier: AEHL_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHL_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 27 704 No. of bytes in distributed program, including test data, etc.: 346 954 Distribution format: tar.gz Programming language: MAPLE 13 and MAPLE 14 Computer: PCs and workstations Operating system: UNIX/LINUX systems and WINDOWS Classification: 4.3 Nature of problem: Determination of analytical properties of systems of differential equations, including symmetry transformations, analytical solutions and conservation laws. Solution method: The package implements in MAPLE some algorithms (discussed in the text) for the study of systems of differential equations. Restrictions: Depends strongly on the system and on the algorithm required. Typical restrictions are related to the solution of a large over-determined system of linear or non-linear differential equations. Running time: Depends strongly on the order, the complexity of the differential system and the object computed. Ranges from seconds to hours.

Algebraic structures and invariant manifolds of differential systems

Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations, since all equations in this class are related to a canonical quadratic differential system: the Lotka–Volterra system. These algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given. Their subalgebras are associated to nonlinear invariant manifolds in the phase space. These manifolds are calculated explicitly. More general algebraic invariant surfaces are also obtained by combining a theorem of Walcher and the Lotka–Volterra canonical form. Applications are given for Lorenz model, Lotka, May–Leonard, and Rikitake systems.

The dynamical system approach to scalar field cosmology

A spatially flat FLRW universe (motivated by inflation) is studied; by a dimensional reduction of the dynamical equations of scalar field cosmology, it is demonstrated that a spatially flat universe cannot exhibit chaotic behaviour. The result holds when the source of gravity is a non-minimally coupled scalar field, for any self-interaction potential and for arbitrary values of the coupling constant with the Ricci curvature. The phase space of the dynamical system is studied, and regions inaccessible to the evolution are found. The topology of the forbidden regions, their dependence on the parameters, the fixed points and their stability character, and the asymptotic behaviour of the solutions are studied. New attractors are found, in addition to those known from the minimal coupling case, certain exact solutions are presented and the implications for inflation are discussed. The equation of state is not prescribed a priori , but rather is deduced self-consistently from the field equations.

Autocorrelation as a source of truncated Lévy flights in foreign exchange rates

We suggest that the ultraslow speed of convergence associated with truncated Levy flights (Phys. Rev. Lett. 73 (1994) 2946) may well be explained by autocorrelations in data. We show how a particular type of autocorrelation generates power laws consistent with a truncated Levy flight. Stock exchanges have been suggested to be modeled by a truncated Levy flight (Nature 376 (1995) 46; Physica A 297 (2001) 509; Econom. Bull. 7 (2002) 1). Here foreign exchange rate data are taken instead. Scaling power laws in the “probability of return to the origin” are shown to emerge for most currencies. A novel approach to measure how distant a process is from a Gaussian regime is presented.

Ergodicity and central-limit theorem in systems with long-range interactions

In this letter we discuss the validity of the ergodicity hypothesis in theories of violent relaxation in long-range interacting systems. We base our reasoning on the Hamiltonian mean-field model and show that the lifetime of quasi-stationary states resulting from the violent relaxation does not allow the system to reach a complete mixed state. We also discuss the applicability of a generalization of the central-limit theorem. In this context, we show that no attractor exists in distribution space for the sum of velocities of a particle other than the Gaussian distribution. The long-range nature of the interaction leads in fact to a new instance of sluggish convergence to a Gaussian distribution.

A numerical method for the stability analysis of quasi-polynomial vector fields

This paper shows the sufficient conditions for the existence of a Lyapunov function in the class of quasi-polynomial dynamical systems. We focus on the cases where the systems parameters are numerically specified. A numerical algorithm to analyze this problem is presented, which involves the resolution of a linear matrix inequality (LMI). This LMI is collapsed to a linear programming problem. From the numerical viewpoint, this computational method is very useful to search for sufficient conditions for the stability of non-linear systems of ODEs. The results of this paper greatly enlarge the scope of applications of a method previously presented by the authors.

Algorithmic complexity theory and the relative efficiency of financial markets

Financial economists usually assess market efficiency in absolute terms. This is to be viewed as a shortcoming. One way of dealing with the relative efficiency of markets is to resort to the efficiency interpretation provided by algorithmic complexity theory. We employ such an approach in order to rank 36 stock exchanges and 20 US dollar exchange rates in terms of their relative efficiency.

Necessary conditions for the existence of quasi-polynomial invariants: the quasi-polynomial and Lotka–Volterra systems

We show that any quasi-polynomial invariant of a quasi-polynomial dynamical system can be transformed into a quasi-polynomial invariant of a homogeneous quadratic Lotka–Volterra dynamical system. We show how this quasi-polynomial invariant can be decomposed in a simple manner. This decomposition permits to conclude that the existence of polynomial semi-invariants in Lotka–Volterra systems is a necessary condition for the existence of quasi-polynomial invariants. We derive a method which allows to construct the necessary conditions for existence of semi-invariants on Lotka–Volterra dynamical systems. Applications are given.

On the Origins of Truncated Levy Flights

Abstract We show that truncated Levy flights appear due to the presence of particular features of autocorrelation in data. We present and analyze ‘physical’ reasons sufficient to ensure the scaling power laws and sluggish convergence associated with truncated Levy flights. Our approach is exemplified with currency data for the British pound and Chinese yuan against the US dollar. We further compare these examples with a simulated Lorentzian distribution.

Stability properties of a general class of nonlinear dynamical systems

We establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems, the so-called quasi-polynomial vector fields, with the help of a natural embedding of such systems in a family of generalized Lotka-Volterra (LV) equations. A purely algebraic procedure is developed to determine such conditions. We apply our method to obtain new results for LV systems, by a reparametrization in time variable, and to study general nonlinear vector fields, originally far from the LV format.