Tham khảo tài liệu 'new approaches in automation and robotics part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes 353 Figure 6. Time series considered for the output of 25 LTV2 Figure 7. Time series considered for the output of 26 A detailed analysis of a portion of the time series obtained from 25 and 26 gives more information on the results obtained through simulation with Mathematica. The time series considered for analysis appear in Figures 6 and 7. In a first attempt Lyapunov exponents may be determined to establish the nature of the behaviour displayed by the systems represented by equations 25 and 26 . However given the periodic character of their time series other methods should be used to assess the stability of 25 and 26 Sprott 2003 . In order to determine whether the stability condition given in 17 is broken or not for equation 25 the power spectra of the time series considered should be obtained. According to Sigeti 1995 the power spectrum of a given signal with positive Lyapunov exponents has an exponential high-frequency falloff relationship. Such characteristic in the frequency domain is due to the fact that the function which defines the signal under consideration has singularities in the complex plane when the time variable t is seen as a complex variable and not as a real one Sigeti 1995 . When the Fourier transform is computed for such a signal the singularities must be avoided in the complex plane through an adequate integration path and in this way exponential terms appear on its associated Fourier transform Sigeti 1995 . In the presence of noise the exponential frequency falloff relationship will be noticeable up to a given frequency and afterwards it will decay as a power of f-n where f is the frequency and n a natural number Lipton Dabke 1996 . These phenomena are also observable in chaotic systems as well independently of the appearance of attractors or not 354 New Approaches in Automation and Robotics in their dynamic behaviour van Wyk . Steeb 1997 . When there are
