Application of the fourth-order three-stage symplectic integrators in chaos determination
School of Science, Nanchang University, 330031, Nanchang, China
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Revised: 26 June 2011
Accepted: 18 July 2011
Published online: 12 August 2011
Global symplectic integrators mean that both the equations of motion and their variational equations of Hamiltonian systems do simultaneously use symplectic integrations. The numerical approaches are very helpful to obtain reliable chaos indicators like Lyapunov exponents and fast Lyapunov indicators, so that correct chaos determination can be ensured. The two new fourth-order three-stage symplectic integrators proposed recently by us are particularly suitable for integrating variational equations of any Hamiltonian in which the kinetic energy is a quadratic form of momenta and the potential energy depends on position coordinates, and also are efficient to solve a class of Hamiltonian systems including the lattice φ 4 models. As a result, both one of the two integrators and the method of fast Lyapunov indicators are recommended to provide some insight into the dynamics of three-dimensional lattice φ 4 models so that a great deal of the computational cost can be saved. A transition to chaos on the variation of single parameter and the structure of associated initial-condition sets for chaotic and regular motions is described.
© Società Italiana di Fisica and Springer, 2011