The lorenz system is a chaotic system that exhibits chaotic behavior
within certain boundaries.
The Rossier (or tornado) pattern is relatively stable on the scope screen for a certain state of chaotic behavior
of the circuit.
Figure 2 indicated that system (1a), (1b), and (1c) can display unbounded orbits and periodic and chaotic behaviors
. In Figure 3, we firstly fix b = 3 and plot the bifurcation diagram with respect to a and the related largest Lyapunov exponent.
Improving the performance of a dynamic system and/or avoiding chaotic behavior
require periodic motion, which is more important when working under specific conditions.
In this figure, the chaotic behavior
in traffic-flow evolution was analyzed under three different types of travel information.
Our case belongs to the more complex chaotic behavior
indicated in (b).
Numerical Chaotic Behavior
of the Fractional Rikitake System World Journal of Modelling and Simulation Vol.
In our case, the value of [beta] is fixed to 3.9999 that corresponds to a highly chaotic case [19, 20], Indeed, the Lyapunov exponent [21, 22] measures the chaotic behavior
of a function and the corresponding Lyapunov exponent of the logistic map for [beta] = 3.9999 is 0.69 very close to its maximum which is 0.59.
In order to verify the chaotic behavior
of the system, Lyapunov exponents diagrams and bifurcation diagrams can be compared together.
If the Melnikov function has a simple zero, then the stable and unstable manifolds intersect transversally, homoclinic bifurcation occurs and hence chaotic behavior
Henceforth, this paper shows, by experiments, that multiscroll chaotic oscillators with optimal MLE can have a more complex chaotic behavior
. In this manner, the optimization task to compute the MLE of a chaotic oscillator based on saturated nonlinear function (SNLF) series is performed herein by applying the differential evolution algorithm already introduced in , and which basically searches for the optimal values of the coefficients of the mathematical description of the multiscroll chaotic oscillator.