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Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems. (English) Zbl 1198.37091

Summary: We study in detail the grid search numerical method to locate symmetric periodic orbits in Hamiltonian systems of two degrees of freedom. The method is based on the classical search method but combining up-to-date numerical algorithms in the search and in the integration process. Instead of using Newton methods that requires to differentiate the Poincaré map we use the Brent’s method and in the integration process a Taylor series method that permits us to compute the orbits using extended precision, something highly interesting in the case of unstable periodic orbits. These facts have permitted us to obtain much more periodic orbits than other researchers. Once the families of periodic orbits have been found we study the bifurcations just by comparing with the stability index and the classical generic bifurcations for Hamiltonian systems with and without symmetries. We illustrate the method with four important classical Hamiltonian problems.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
70-08Computational methods (mechanics of particles and systems)
70H05Hamilton’s equations
70H12Periodic and almost periodic solutions (mechanics of particles and systems)