Abstract
Fractional calculus is a topic being more than 300 years old. The idea of fractional calculus has been known since the regular calculus, with the first reference probably being associated with Leibniz and L’Hospital in 1695 where half-order derivative was mentioned. In a correspondence between Johann Bernoulli and Leibniz in 1695, Leibniz mentioned the derivative of general order. In 1730 the subject of fractional calculus did not escape Euler’s attention. J. L. Lagrange in 1772 contributed to fractional calculus indirectly, when he developed the law of exponents for differential operators. In 1812, P. S. Laplace defined the fractional derivative by means of integral and in 1819 S. F. Lacroix mentioned a derivative of arbitrary order in his 700-page long text, followed by J. B. J. Fourier in 1822, who mentioned the derivative of arbitrary order. The first use of fractional operations was made by N. H. Abel in 1823 in the solution of tautochrome problem. J. Liouville made the first major study of fractional calculus in 1832, where he applied his definitions to problems in theory. In 1867, A. K. Grünwald worked on the fractional operations. G. F. B. Riemann developed the theory of fractional integration during his school days and published his paper in 1892. A. V. Letnikov wrote several papers on this topic from 1868 to 1872. Oliver Heaviside published a collection of papers in 1892, where he showed the so-called Heaviside operational calculus concerned with linear generalized operators. In the period of 1900 to 1970 the principal contributors to the subject of fractional calculus were, for example, H. H. Hardy, S. Samko, H. Weyl, M. Riesz, S. Blair, etc.
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References
Ahmad W. M., 2005, Hyperchaos in fractional order nonlinear systems, Chaos, Solitons and Fractals, 26, 1459–1465.
Andrievskii B. R. and Fradkov A. L., 2003, Control of chaos: Methods and applications. I. Methods, Automation and Remote Control, 64, 673–713.
Andrievskii B. R. and Fradkov A. L., 2004, Control of chaos: Methods and applications. II. Applications, Automation and Remote Control, 65, 505–533.
Arena P., Caponetto R., Fortuna L. and Porto D., 1998, Bifurcation and chaos in noninteger order cellular neural networks, International Journal of Bifurcation and Chaos, 8, 1527–1539.
Arena P., Caponetto R., Fortuna L. and Porto D., 2000, Nonlinear Noninteger Order Circuits and Systems — An Introduction, World Scientific, Singapore.
Axtell M. and Bise E. M., 1990, Fractional calculus applications in control systems, Proc. of the IEEE Nat. Aerospace and Electronics Conf., New York, 563–566.
Bode H. W., 1949, Network Analysis and Feedback Amplifier Design, Tung Hwa Book Company, Shanghai.
Cafagna D., 2007, Fractional Calculus: A mathematical tool from the past for present engineers, IEEE Industrial Electronics Magazine, 1, 35–40.
Carlson G. E. and Halijak C. A., 1964, Approximation of fractional capacitors (1/s)1/n by a regular Newton process, IEEE Trans. on Circuit Theory, 11, 210–213.
Charef A., Sun H. H., Tsao Y. Y. and Onaral B., 1992, Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, 37, 1465–1470.
Deng W., Li C. and Lu J., 2007, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynamics, 48, 409–416.
Dorcák Ľ, 1994, Numerical Models for the Simulation of the Fractional-Order Control Systems, UEF-04-94, The Academy of Sciences, Inst. of Experimental Physic, Košice, Slovakia.
Gao X. and Yu J., 2005, Chaos in the fractional order periodically forced complex Duffing’s oscillators, Chaos, Solitons and Fractals, 24, 1097–1104.
Guo L. J., 2005, Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems, Chinese Physics, 14, 1517–1521.
Hartley T. T., Lorenzo C. F. and Qammer H. K., 1995, Chaos on a fractional Chua’s system, IEEE Trans. Circ. Syst. Fund. Theor. Appl., 42, 485–490.
Li C. and Chen G., 2004, Chaos and hyperchaos in the fractional-order Rossler equations, Physica A, 341, 55–61.
Lu J. G., 2005a, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos, Solitons and Fractals, 26, 1125–1133.
Lu J. G., 2005b, Chaotic dynamics and synchronization of fractional-order Chua’s circuits with a piecewise-linear nonlinearity, Int. Journal of Modern Physics B, 19, 3249–3259.
Magin R. L., 2006, Fractional Calculus in Bioengineering, Begell House Publishers, Redding.
da Graca Marcos, M., Duarte F. B. M. and Machado J. A. T., 2008, Fractional dynamics in the trajectory control of redundant manipulators, Communications in Nonlinear Science and Numerical Simulations, 13, 1836–1844.
Miller K. S. and Ross B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons. Inc., New York.
Nakagava M. and Sorimachi K., 1992, Basic characteristics of a fractance device, IEICE Trans. fundamentals, E75-A, 1814–1818.
Nimmo S. and Evans A. K., 1999, The effects of continuously varying the fractional differential order of chaotic nonlinear systems, Chaos, Solitons and Fractals, 10, 1111–1118.
Oldham K. B. and Spanier J., 1974, The Fractional Calculus, Academic Press, New York.
Oustaloup A., 1995, La Derivation Non Entiere: Theorie, Synthese et Applications, Hermes, Paris.
Parada F. J. V., Tapia J. A. O. and Ramirez J. A., 2007, Effective medium equations for fractional Fick’s law in porous media, Physica A, 373, 339–353.
Podlubny I., 1999a, Fractional Differential Equations, Academic Press, San Diego.
Podlubny I., 1999b, Fractional-order systems and PI λ D μ-controllers, IEEE Transactions on Automatic Control, 44, 208–213.
Silva C. P., 1993, Shil’nikov’s theorem — A tutorial, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 40, 675–682.
Tavazoei M. S. and Haeri M., 2007, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems, IET Signal Proc., 1, 171–181.
Tavazoei M. S. and Haeri M., 2008, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Analysis, 69, 1299–1320.
Torvik P. J. and Bagley R. L., 1984, On the appearance of the fractional derivative in the behavior of real materials, Transactions of the ASME, 51, 294–298.
Tseng C. C., 2007, Design of FIR and IIR fractional order Simpson digital integrators, Signal Processing, 87, 1045–1057.
Vinagre B. M., Chen Y. Q. and Petráš I., 2003, Two direct Tustin discretization methods for fractional-order differentiator/integrator, J. Franklin Inst., 340, 349–362.
Wang J. C., 1987, Realizations of generalized Warburg impedance with RC ladder networks and transmission lines, Journal of The Electrochemical Society, 134, 1915–1920.
West B. J., Bologna M. and Grigolini P., 2002, Physics of Fractal Operators, Springer, New York.
Westerlund S., 2002, Dead Matter Has Memory!, Causal Consulting, Kalmar, Sweden.
Zaslavsky G. M., 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
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Petráš, I. (2011). Introduction. In: Fractional-Order Nonlinear Systems. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18101-6_1
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