Yorke, "
Chaotic behavior of multidimensional difference equations," in Functional Differential Equations and Approximation of Fixed Points (Proc.
However, in the case of chaos occurrence, the complexity of
chaotic behavior is relatively small under real-time information alone.
Due to the frequent use of feedback systems and all tools that exist for its analysis, it has been studied under which structural conditions a
chaotic behavior is presented.
In particular, this new topology exhibits
chaotic behavior, with a great complexity, and develops an attractor (see Figure 21) very similar to that shown in Figure 13(a), although with a higher Kaplan-Yorke dimension, with a value of [D.sub.KY] = 2.52.
Also, the used parameter [lambda] (equal to 4) does not provide the best
chaotic behavior. In our case, combining three chaotic logistic maps allows to increase the length of the global resulting cycle, which is given by the LCM of the three cycle lengths.
More precisely, this condition provides a domain on the parameter spaces where the system has transverse homoclinic orbits resulting in possible
chaotic behavior. As it can be observed from this figure, as [mu] increases the threshold [f.sub.p] obtained by the Melnikov method increases on the frequency 0 < [omega] < 1.5.
The experiments show that the
chaotic behavior is more complex when the MLE is optimized.
In the absence of the controller, the stochastic Chen system (17) exhibits a
chaotic behavior as shown in Figure 1.
Figures 1(c) and 1(d) show that the neurons exhibit
chaotic behavior when the largest Lyapunov exponent becomes greater than 0.
The results confirmed presence of
chaotic behavior in the Iranian equity market.
To observe the system dynamics, I consider [[tau].sub.1] = 0.56 and [[tau].sub.2] = 0.535, both are beyond their stability range, and then the system shows
chaotic behavior (Figure 6).
In this section, the
chaotic behavior of model (1) with the parameters G, b, a, and F are discussed, and the complex dynamic behaviors are analyzed by Lyapunov exponents spectrum, bifurcation diagram, Poincare section, and power spectrum.
