419 research outputs found
Explicit representations of biorthogonal polynomials
Given a parametrised weight function such that the quotients
of its consecutive moments are M\"obius maps, it is possible to express the
underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present
paper we address ourselves to two related issues. Firstly, we demonstrate that,
subject to additional assumptions, every such obeys (in ) a linear
differential equation whose solution is a generalized hypergeometric function.
Secondly, using a generalization of standard divided differences, we present a
new explicit representation of the underlying orthogonal polynomials
A new simulation technique for RF oscillators
The study of phase-noise in oscillators and the design of new circuit topologies necessitates an efficient technique for the simulation of oscillators. While numerous approaches have been developed over the years e.g. [1-3], each has its own merits and demerits. In this contribution, an asymptotic numeric method developed in e.g. [4-5] is applied to the simulation of RF oscillators. The method is closely related to the stroboscopic and high-order averaging method in [6] and the Heterogeneous Multiscale Methods in [7]. The method is advantageous in that the same methodology can be applied for the simulation of general circuit problems involving highly oscillatory ordinary differential equations, partial differential equations and delay differential equations. Furthermore and counter-intuitively, its efficacy improves with increasing frequency, a feature that is very favourable in modern communications systems where operating frequencies are ever rising. Results for a CMOS oscillator will confirm the validity and efficiency of the proposed method
The asymptotic behaviour of certain difference equations with proportional delays
AbstractThis paper is concerned with the nonstationary linear difference equation yn = ÎŽynâ1 + ÎŒ(y[n/2] + y[(nâ1)/2]), y0 = 1. We demonstrate by Fourier techniques that the sequence is asymptotically stable when |λ| < 1 and |ÎŒ| < (12)|1 â λ|, but our main effort is devoted to the marginal case |λ| = 1. We derive the solution explicitly as a power series in ÎŒ for λ = â1M, thereby demonstrating that it is uniformly bounded and that, for ÎŒâ 0, its attractor contains a countable subset of distinct points. Finally, we consider a somewhat more general difference equation and prove that, for specific choices of parameters therein, the attractor is a probabilistic mixture of Julia sets
The Pantograph Equation in the Complex Plane
AbstractThe subject matter of this paper focuses on two functional differential equations with complex lag functions. We address ourselves to the existence and uniqueness of solutions and to their asymptotic behaviour
A Unified Approach to Spurious Solutions Introduced by Time Discretisation. Part I: Basic Theory
The asymptotic states of numerical methods for initial value problems are examined. In particular, spurious steady solutions, solutions with period 2 in the timestep, and spurious invariant curves are studied. A numerical method is considered as a dynamical system parameterised by the timestep h. It is shown that the three kinds of spurious solutions can bifurcate from genuine steady solutions of the numerical method (which are inherited from the differential equation) as h is varied. Conditions under which these bifurcations occur are derived for RungeâKutta schemes, linear multistep methods, and a class of predictor-corrector methods in a PE(CE)^M implementation. The results are used to provide a unifying framework to various scattered results on spurious solutions which already exist in the literature. Furthermore, the implications for choice of numerical scheme are studied. In numerical simulation it is desirable to minimise the effect of spurious solutions. Classes of methods with desirable dynamical properties are described and evaluated
An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions
We study the efficient approximation of integrals involving Hankel functions
of the first kind which arise in wave scattering problems on straight or convex
polygonal boundaries. Filon methods have proved to be an effective way to
approximate many types of highly oscillatory integrals, however finding such
methods for integrals that involve non-linear oscillators and
frequency-dependent singularities is subject to a significant amount of ongoing
research. In this work, we demonstrate how Filon methods can be constructed for
a class of integrals involving a Hankel function of the first kind. These
methods allow the numerical approximation of the integral at uniform cost even
when the frequency is large. In constructing these Filon methods we
also provide a stable algorithm for computing the Chebyshev moments of the
integral based on duality to spectral methods applied to a version of Bessel's
equation. Our design for this algorithm has significant potential for further
generalisations that would allow Filon methods to be constructed for a wide
range of integrals involving special functions. These new extended Filon
methods combine many favourable properties, including robustness in regard to
the regularity of the integrand and fast approximation for large frequencies.
As a consequence, they are of specific relevance to applications in wave
scattering, and we show how they may be used in practice to assemble
collocation matrices for wavelet-based collocation methods and for hybrid
oscillatory approximation spaces in high-frequency wave scattering problems on
convex polygonal shapes
Numerical Implementation of Gradient Algorithms
A numerical method for computational implementation of gradient dynamical systems is presented. The method is based upon the development of geometric integration numerical methods, which aim at preserving the dynamical properties of the original ordinary differential
equation under discretization. In particular, the proposed method belongs to the class of discrete gradients methods, which substitute the gradient of the continuous equation with a discrete gradient, leading to a map that possesses the same Lyapunov function of the dynamical system,
thus preserving the qualitative properties regardless of the step size. In this work, we apply a discrete gradient method to the implementation of Hopfield neural networks. Contrary to most geometric integration
methods, the proposed algorithm can be rewritten in explicit form, which considerably improves its performance and stability. Simulation results show that the preservation of the Lyapunov function leads to an improved performance, compared to the conventional discretization.Spanish Government project no. TIN2010-16556 Junta de AndalucĂa project no. P08-TIC-04026 Agencia Española de CooperaciĂłn Internacional
para el Desarrollo project no. A2/038418/1
Symmetry Reduction of Optimal Control Systems and Principal Connections
This paper explores the role of symmetries and reduction in nonlinear control
and optimal control systems. The focus of the paper is to give a geometric
framework of symmetry reduction of optimal control systems as well as to show
how to obtain explicit expressions of the reduced system by exploiting the
geometry. In particular, we show how to obtain a principal connection to be
used in the reduction for various choices of symmetry groups, as opposed to
assuming such a principal connection is given or choosing a particular symmetry
group to simplify the setting. Our result synthesizes some previous works on
symmetry reduction of nonlinear control and optimal control systems. Affine and
kinematic optimal control systems are of particular interest: We explicitly
work out the details for such systems and also show a few examples of symmetry
reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure
- âŠ