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In this paper, we introduce the notions of the almost anti-periodic discrete process of the N-dimensional vector-valued and N×N matrix-valued functions. Some basic properties of the almost anti-periodic discrete functions are established. Based on this, the conditions of the stability and instability of the almost anti-periodic solutions to the general N-dimensional mechanical system and the underactuated Euler–Lagrange system have been considered. Moreover, some examples are provided to support our obtained results.

Almost Anti-Periodic Discrete Oscillation of General N-Dimensional Mechanical System and Underactuated Euler-Lagrange System