# chaos theory

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## chaos theory

a branch of mathematics that seeks to predict widespread effects of small (or minute) and possibly remote triggering events; the unpredictable course of some epidemics and malignant metastases may accord with chaos theory.

## chaos theory

The mathematical conception that some phenomena that seem random may be of a deterministic order highly sensitive to initial conditions and perturbations. There is a growing appreciation that chaos may be a feature of many biological systems and that chaos theory may prove to have many applications in medicine.
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The lorenz system is a chaotic system that exhibits chaotic behavior within certain boundaries.
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.
In this work the chaotic behavior of a feedback system proposed is evaluated using Lyapunov exponents and phase plane.
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 is expected.
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 [8], and which basically searches for the optimal values of the coefficients of the mathematical description of the multiscroll chaotic oscillator.

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