On the behavior of Linear Dependence, Smaller, and Generalized Alignment Indices in discrete and continuous chaotic systems

The discrimination between order and chaos in dynamical systems remains a central problem in the field. Among the most widely used indicators are the Smaller Alignment Index (SALI), the Generalized Alignment Index (GALI), and the Linear Dependence Index (LDI), all of which exploit the evolution of deviation vectors to distinguish regular from chaotic motion. In this paper, we first show analytically, and confirm numerically, that the decay rates of LDI for chaotic orbits in both discrete- and continuous-time systems are the same with those of GALI reported in the literature. Our derivations, however, are more accessible, relying on the Singular Value Decomposition rather than the wedge-product formulation of GALI, which involves volumes of higher-dimensional parallelepipeds. We then derive the analytical expression for the decay rate of SALI in chaotic maps, demonstrating that it depends on the difference of the two largest Lyapunov exponents, as previously established for continuous-time systems. Crucially, we show analytically that the second Lyapunov exponent must always be considered, independent of its sign, in order to capture correctly the decay of SALI. This contrasts with existing results for continuous systems, where the second exponent is greater or equal than zero for chaotic orbits. Our analytical and numerical findings, therefore, extend the SALI decay rate formula to both continuous- and discrete-time systems. Finally, we confirm numerically that the decay rate of the SALI for chaotic maps is accurately described by our formula, which incorporates the two largest Lyapunov exponents, regardless of whether the second exponent is positive, zero, or negative.