We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.
This paper introduces the basic principles and calculation methods for the correlation dimension and Kolmogorov entropy. By calculating the correlation dimension and Kolmogorov entropy when the gear is under different working conditions, we can analyze the inherent relationship between the two in depicting of the running condition of the gearbox. The result shows that,the correlation dimension and Kolmogorov entropy have a good consistency in the description of working status of gearbox. This conclusion not only provides a good basis for the gearbox running condition judgment and fault diagnosing, but can also provide the experimental basis for the chaotic characteristic parameters selection in state monitoring and fault diagnosing.
Functions are derived, which are orthonormal on the range r=0, 1, with weight function
corresponding to the distribution of r in a typical experimental procedure for measurement
of the two-point orientation–coherence (or orientation–correlation) function.
These are obtained by making an appropriate change of variable in spherical Bessel
functions, orthonormal on the range r=0, 1, with unit weight function. The effects of
weight function and change of variable on the functions are considered.
r-Dependence in Two-Point Orientation Coherence Functions