Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations

Abstract

Abstract The largest Lyapunov exponent (LLE) is an important tool used to identify systems. However, it cannot easily distinguish between chaos and hyperchaos. Therefore, the second-largest Lyapunov exponent (SLLE) must be calculated. Some methods to calculate this index have already been proposed, but these require massive computation times and data points. Thus, to reduce the calculating complexity, we propose a simple novel method based on three nearby orbits to directly calculate the finite-time local SLLE from continuous chaotic systems with improved accuracy. First, we obtain a solution orbit of chaotic equations. Then, two selected points from the solution are used as the initial condition to solve the same equation and obtain two solution orbits. Next, we calculate the evolution area of three solution orbits and the distance between the two solution trajectories. The LLE and area exponent are then obtained from the logarithmic curve of the track distance and the area. Finally, the SLLE is obtained from the difference computed between the two indices. Some chaotic differential equations and non-chaotic systems are presented to demonstrate the efficiency of the proposed method. Calculating the finite-time local exponents does not require the calculation of all the evolution lengths. Thus, the proposed method is simple, very fast and robust without the need to reconstruct the phase space. For now, the findings of this research provide new ideas related to the SLLE calculation theory.