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Sensitivity tools vs. Poincaré sections. (English) Zbl 1092.37531
Summary: We introduce a modification of the fast Lyapunov indicator (FLI) denominated ${\text{OFLI}}_{\text{TT}}^{2}$ indicator that may provide a global picture of the evolution of a dynamical system. Therefore, it gives an alternative or a complement to the pictures given by the classical Poincaré sections and, besides, it may be used for any dimension. We present several examples comparing with the Poincaré sections in two classical problems, the Hénon-Heiles and the extensible-pendulum problems. Besides, we show the application to Hamiltonians of three degrees of freedom as an isotropic harmonic oscillator in three dimensions perturbed by a cubic potential and non-Hamiltonian problems as a four-dimensional chaotic system. Finally, a numerical method especially designed for its computation is presented in the appendix.

MSC:
 37N05 Dynamical systems in classical and celestial mechanics 37D45 Strange attractors, chaotic dynamics 70K55 Transition to stochasticity (chaotic behavior) 70H14 Stability problems (mechanics of particles and systems)