Linear ordinary differential equations (ODEs) with periodic coefficients appear in various interesting applications, such as determining the linear stability regions of systems of vertically driven multiple pendula. Sinha and Butcher [1
] have obtained very good approximations to the solutions of such equations by calculating approximate Picard iterates symbolically in the parameters on which the system depends. In this article we show an improvement to the method of Sinha and Butcher. We are able to calculate exact, rather then approximate, Picard iterates of high order. The key point in the programming is the necessity of introducing a user-defined function to carry out the integrations that appear in the definition of the Picard iterates. After introducing the concept of Picard iteration and explaining its fast implementation, we apply the method to determine the stability regions for linearized systems of vertically driven multiple pendula.
2. Picard Iterates
3. Picard Iteration for Systems of Linear ODEs with Periodic Coefficients
4. Fast Implementation of Picard Iteration for Systems of Linear ODEs with Periodic Coefficients and the FastPicard Package
5. Application to Linearized Systems of Driven Pendula
About the Author
Armando G. M. Neves majored in physics in 1986 at the Federal University of Minas Gerais (UFMG) in Brazil. After receiving a master's degree, he obtained his doctoral degree in physics at Rome University "La Sapienza" in Italy in 1993. Although his degrees are in physics, he has always searched for mathematical rigor. Since 1992 he has held a position in the Mathematics Department of UFMG.
His general research interest is mathematical physics. His principal interests are in statistical mechanics, quantum field theory, and, more recently, dynamical systems. Although his first academic works were noncomputational, after using Mathematica he realized that symbolic computation opened up entirely new fields of problems and provided new ways of attacking older problems.
Armando G. M. Neves
UFMG - Departamento de Matemática
Av. Antônio Carlos, 6627 - Caixa Postal 702
30123-970 Belo Horizonte - MG