Computer Algebra System MACSYMA Research Articles

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.

Identities Arising From Hecke Transformations of Modular Forms Over Q( [formula omitted]) and Q( [formula omitted

The Hecke transform is used on Hilbert modular forms over Q( 2 ) and Q ( 3 ) to produce unusual numerical identities. The case for Q( 2 ) is complicated by the fact that there are two spaces of modular forms. The computer algebra system MACSYMA was used to find eigenfunctions of the Hecke transform, and to calculate identities involving the analogue of the sum of divisors functions.

Free vibrations of a thin elastica by normal modes

In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.

A theoretical solution for the buckling of sandwich panels with laminated face plates using a computer algebra system

Abstract The governing equations for both symmetric and asymmetric wrinkling of sandwich panels are derived using operator notation. In each case differential systems of three equations in three unknowns are obtained and two of the three variables are eliminated by taking combinations of the equations. Whilst the mathematical techniques employed are straightforward, the expressions involved are lengthy and recourse is made to the computer algebra system MACSYMA, without which a solution for the asymmetric case would have been difficult to achieve. Explicit algebraic expressions are given for buckling loads which can be used by structural engineers to calculate allowable design loads for sandwich panels.

Macsyma computation of local minimal realization of dynamical systems of which generating power series are finite

We present here a package of Macsyma programs, allowing the manipulation of words, and noncommutative power series over some finite alphabet. On the basis of the works of M.Fliess and C.Reutenauer, concerning local realization of nonlinear dynamical systems, we present an algorithm allowing the computation of the local and minimal realization of finite generating power series. We describe that algorithm in the computer algebra system Macsyma.

An analytical approach to global optimization

Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local optimum is found, or if a global optimum is detected no proof is provided that it is one. We study here the extent to which such global optimization problems can be solved exactly using analytical methods. To this effect, we propose a series of tests, similar to those of combinatorial optimization, organized in a branch-and-bound framework. The first complete solution of two difficult test problems illustrates the efficiency of the resulting algorithm. Computational experience with the programbagop, which uses the computer algebra systemmacsyma, is reported on. Many test problems from the compendiums of Hock and Schittkowski and others sources have been solved.

Averaging using elliptic functions: approximation of limit cycles

We apply the method of averaging to first order in the small parameter e to the autonomous system $$x'' + \alpha x + \beta x^3 + \varepsilon g\left( {x, x'} \right) = 0$$ where we do not consider β as small. This involves perturbing off of Jacobian elliptic functions, rather than off of trigonometric functions as is usually done. The resulting equations involve integrals of elliptic functions which are evaluated using a program written in the computer algebra system MACSYMA. The results are applied to the problem of approximating limit cycles in the above differential equation.

An Automated Procedure for Globally Optimal Design

An experimental computer system for globally optimal design, called BAGOP, is discussed. This new tool uses the computer algebra system MACSYMA to implement a variety of tests and branching rules in a flexible branch-and-bound framework. Many problems of globally optimal design have been solved by BAGOP, some of them for the first time, and most of them for the first time in an entirely automated way.

Normal forms and periodic solutions in the theory of non-linear oscillations. Existence and asymptotic theory

A concept of normal forms is presented to analyse a class of O.D.E.'s arising from the field of non-linear oscillations. An algorithm for the calculation of the normalized differential equations is presented as well as theorems on the existence, uniqueness and stability of periodic solutions and their calculation in an approximative way by using the normalized differential equations. As an application of the theory a simple model is discussed describing the flow-induced vibrations of an oscillator with one degree of freedom in a uniform windfield. The necessary calculations to perform the analysis have been done by an implementation of the algorithms for the calculation of higher order normal forms, as presented here, on the Computer-Algebra system Macsyma.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.