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Lyapunov exponents of a class of piecewise continuous systems of fractional order

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Abstract

In this paper, we prove that a class of piecewise continuous autonomous systems of fractional order has well-defined Lyapunov exponents. To do so, based on some known results from differential inclusions of integer order and fractional order, as well as differential equations with discontinuous right-hand sides, the corresponding discontinuous initial value problem is approximated by a continuous one with fractional order. Then, the Lyapunov exponents are numerically determined using, for example, Wolf’s algorithm. Three examples of piecewise continuous chaotic systems of fractional order are simulated and analyzed: Sprott’s system, Chen’s system, and Simizu–Morioka’s system.

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Notes

  1. Actually, in the great majority of known examples, with \(g\) being polynomial, it is also a smooth function.

  2. This class of sigmoid functions includes many other examples such as the hyperbolic tangent, the error function, the logistic function, algebraic functions like \(\frac{x}{\sqrt{\delta +x^2}}\), \(\frac{2}{1+e^{-\frac{x}{\delta }}}-1\) [13] and so on.

  3. Because, while the problem is solved in parallel with (13), the initial conditions change, so the usual notation \(x_0\) is replaced with \(x\).

  4. In this paper, the Matlab code fde12.m [40] (an implementation of the predictor-corrector PECE method of Adams–Bashforth–Moulton type presented in [31], which has been suitably modified for the general case with the incommensurate orders) has been utilized.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Avram Iancu University, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute for Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

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Danca, MF. Lyapunov exponents of a class of piecewise continuous systems of fractional order. Nonlinear Dyn 81, 227–237 (2015). https://doi.org/10.1007/s11071-015-1984-6

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