The Lindstedt-Poincaré Technique as an Algorithm for Computing Periodic Orbits

• Published in: SIAM review Vol. 43; no. 3; pp. 478 - 495
• Main Author: Viswanath, Divakar
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.09.2001
• Subjects: Algorithms; Approximation; Eigenvalues; Exact sciences and technology; Fourier series; General topology; Limit cycles; Lunar orbits; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Orbits; Ordinary differential equations; Periodic orbits; Problems and Techniques; Sciences and techniques of general use; Theory; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Trajectories; Vector fields; Perturbation theory; Differential equation; Josephson junction; Periodic orbit; Perturbation; Dynamical system; Computing; Algorithm; Equation system; Kepler motion; Hill problem; Linear system; Linstedt Poincaré technique; Perturbation techniques; Orbit determination; Facility; Periodic system
• ISSN: 0036-1445; 1095-7200
• DOI: 10.1137/S0036144500375292

The Lindstedt-Poincaré technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations. It uses a nearby periodic orbit of the unperturbed differential equation as the first approximation. We derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems. The algorithm, unlike the Lindstedt-Poincaré technique, does not require the dynamical system to be a small perturbation of a solvable differential equation. This makes it more broadly applicable. The algorithm is quadratically convergent. It works with equal facility, as examples show, irrespective of whether the periodic orbit is attracting, or repelling, or a saddle. One of the examples presents what is possibly the most accurate computation of Hill's orbit of lunation since its justly celebrated discovery in 1878.