Determining accurate Lyapunov exponents of a multiscroll chaotic attractor based on SNFS

In n-scroll attractors design using active devices, the behavior of the nonlinear part is, often, modeled by using a piecewise linear (PWL) approach. Although little effort is required to build a PWL model, the main performance parameters associated with the active device to be used in the physical design are not included. Moreover, the main metrics used for quantifying chaos are: Lyapunov exponent (LE), Kolmogorov-Sinai entropy, Kaplan-Yorke dimension, fractal power spectra and fractal dimension. Among them, the former not only is used as indicator of chaotic motion when a positive LE is found, but if its value positively increases, then the unpredictability grade of the chaotic system also increases. All reported works until today about the computed LEs on chaotic systems are derived using a PWL approach. This is a serious drawback since PWL models cannot model a dynamic behavior, and hence, introducing a level of error during the numerical analysis becomes evident when experimental and numerical results are examined at high frequency. Under this scheme, new results on the accurate computing of LEs for a chaotic oscillator based on a nonlinear function called saturated nonlinear function series (SNFS) are introduced. A mathematical model is used for characterizing the dynamical behavior of the SNFS that includes the main performance parameters of Op Amps such as slew rate, DC gain, gain bandwidth product and dynamic range, showing a much better accuracy than the traditional PWL approach. This behavioral modeling demonstrates that the positive LE increases in relation to the number of scrolls and even is also depending on the frequency of the system, overthrowing the results that have been handled in previous works, where a constant value of the LE was obtained when the number of scrolls increased.