Chaos in Nonlinear Fractional Systems

Chaos in Nonlinear Fractional Systems

Nasr-eddine Hamri (University of Mila, Algeria)
DOI: 10.4018/978-1-5225-5418-9.ch012

Abstract

The first steps of the theory of fractional calculus and some applications traced back to the first half of the nineteenth century, the subject only really came to life over the last few decades. A particular feature is that fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional models in comparison with classical integer-order models; another feature is that scientists have developed new models that involve fractional differential equations in mechanics, electrical engineering. Many scientists have become aware of the potential use of chaotic dynamics in engineering applications. With the development of the fractional-order algorithm, the dynamics of fractional order systems have received much attention. Chaos cannot occur in continuous integer order systems of total order less than three due to the Poincare-Bendixon theorem. It has been shown that many fractional-order dynamical systems behave chaotically with total order less than three.
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Fractional Calculus Fundamentals

We introduce some necessary mathematical tools that will arise in the study of the concepts of fractional calculus. These are the Laplace transform, the Gamma function, the Beta function and the Mittag-Leffler Function.

The Laplace Transform

The Laplace transform is a powerful tool that we shall exploit in investigation of fractional differential equations. We denote the Laplace transform of a function by the symbol , or when convenient, by . More detailed information may be found in (Ditkin, 1966; Doetsch, 1974).

The Laplace transform of a function of a real variable t+is formally defined by

(1)

If the integral in (1) is convergent at s0ℂ, then it converges absolutely for sℂ such that.

The inverse Laplace transform is given for t+by the formula

(2)

Obviously, and are linear integral operators. The direct and the inverse Laplace transforms are inverse to each other for “sufficiently good” functions f and g

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