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The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits. (English) Zbl 0979.65115

Summary: 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.

MSC:
65P10Numerical methods for Hamiltonian systems including symplectic integrators
37C27Periodic orbits of vector fields and flows
37M15Symplectic integrators (dynamical systems)