Lyapunov exponent of nonlinear system of two stage planetary gear system

Lyapunov exponent represents the average exponential divergence rate of the distance between two adjacent trajectories in the phase plane with time, which is a quantitative index of chaotic state. In order to calculate the Lyapunov exponent, the method proposed by Wolf is used, and the re orthogonal vector is obtained by gram Schmidt renormalization.

A “datum” trajectory is defined by the initial condition, and an initial orthonormal principal axis vector coordinate system {E1, E2,… Is attached to the trajectory Where EI is the orthonormal basis. New vector set {V1, V1 Since each vector of chaotic system tends to decline along the direction of local fastest growth, it needs to be orthogonalized when the size and direction deviation is too large:

Where, λ J is the degree of dispersion of the nearby trajectories.

The spectrum of Lyapunov exponent is arranged in order, that is, λ 1 > λ 2 > >λn。 The largest Lyapunov exponent λ Max is usually used as the symbol of chaotic motion.

Because of the piecewise function due to the gap, the Jacobi matrix does not exist everywhere in the conventional sense, so the proposed method can obtain the maximum Lyapunov exponent of the system.