Preservation of Lyapunov stability through effective discretization in Runge–Kutta method

To analyze continuous-time dynamic systems, it is often necessary to discretize them. Traditionally, this has been accomplished using various variants of the Runge–Kutta (RK) method and other available discretization schemes. However, recent advancements have revealed that effective discretization can be achieved by considering the precision of the computer. In studying the stability of such continuous systems according to Lyapunov theory, it is imperative to consider the Lyapunov function of dynamic systems described by differential equations, as well as their discrete counterparts. This study demonstrates that the discretization using the RK method and the effective discretization based on the reduced Runge–Kutta (RRK) method, wherein terms are reduced due to computational precision, preserve the Lyapunov stability across different step-size values. Despite a notable reduction in the number of terms, particularly evident in the fourth-order Runge–Kutta method, stability according to Lyapunov remains intact. Furthermore, reducing the number of terms decreases the operations required at each iteration, yielding reductions of up to 46.67%, 93.58%, and 99.91% for RRK2, RRK3, and RRK4, respectively, in the numerical example. This directly impacts computational cost, as illustrated in the numerical experiments.