New Chaos Indicators for Systems with Extremely Small Lyapunov Exponents

A new chaos indicator Ultra Fast Lyapunov Indicator (UFLI) is proposed in this paper. UFLI uses second order derivatives and can distinguish chaotic orbits from torus orbits (orbits on a torus) more sensitively than FLI, OFLI and OFLI² in a short iterations for the standard map for δ = 0.9 and the Froeschlé map ε = 0.6 if we chose an initial variational vector properly. For the generalized Boole transformations (Boole transformation) whose analytic formula of Lyapunov exponent is given, the value of UFLI grows more rapidly than FLI except for α = 0.999, which is the weak chaos near to the infinite ergodic point while the difference in growing speed of the indicator values is relatively small compared with the standard map or the Froeschlé map.