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[en] In this paper, k step chaometry (k SCM) is defined based on monopolized sphere and instantaneous chaometry (ICM), and the convergent theorem of asymptotical periodic orbit is also proved. The 400 SCM of the discrete model of Lorenz system is calculated and results disclose that 400 SCM can clearly identify the parameters of chaotic dynamic system. The EEG instrument is applied to measure time series of EEG, and it is observed that the instantaneous chaometry of the EEG and the data generated from Lorenz attractor produce similar results.
[en] The concepts of uniform index and expectation uniform index are two mathematical descriptions of the uniformity and the mean uniformity of a finite set in a polyhedron. The concepts of instantaneous chaometry (ICM) and k step chaometry (k SCM) are introduced in order to apply the method in statistics for studying the nonlinear difference equations. It is found that k step chaometry is an indirect estimation of the expectation uniform index. The simulation illustrate that the expectation uniform index for the Lorenz System is increasing linearly, but increasing nonlinearly for the Chen's System with parameter b. In other words, the orbits for each system become more and more uniform with parameter b increasing. Finally, a conjecture is also brought forward, which implies that chaos can be interpreted by its orbit's mean uniformity described by the expectation uniform index and indirectly estimated by k SCM. The k SCM of the heart rate showes the feeble and old process of the heart.
[en] Uniform index is a conception that can describe the uniformity of a finite point set in a polyhedron, and is closely related to chaos. In order to study uniform index, the concept of contained uniform index is defined, which is similar to uniform index and has good mathematical properties. In this paper, we prove the convergence of the contained uniform index, and develop the base of proving the convergence of uniform index.