Page 618 - Mathematical Techniques of Fractional Order Systems

588 Mathematical Techniques of Fractional Order Systems

the same purpose. Analytical results for both stabilizing and synchronizing
controllers are derived while using a systematic backstepping procedure
along with Mittag Leffler and Lyapunov stability results. The proposed
approach ensures global stability and asymptotic synchronization which is
evident from the simulation results presented at the end. The results obtained
here can also be used to address the problem of secure communication and
image encryption. The future research can target the control and synchroni-
zation of fractional order systems with uncertain parameters. The implemen-
tation of these outcomes on software and finally on hardware is the
challenge which can act as the driving force for upcoming researchers in the
area of fractional order chaotic systems. Moreover, stability of fractional
order systems and its interpretation is still an issue and has to be answered.

REFERENCES
Aghababa, M.P., 2012. Robust stabilization and synchronization of a class of fractional-order
chaotic systems via a novel fractional sliding mode controller. Commun. Nonlinear Sci.
Numer. Simul. 17 (6), 2670 2681.
Agrawal, S.K., Srivastava, M., Das, S., 2012. Synchronization of fractional order chaotic systems
using active control method. Chaos Solitons Fractals 45 (6), 737 752.
Aguila-Camacho, N., Duarte-Mermoud, M., Gallegos, J., 2014. Lyapunov functions for frac-
tional order systems. Commun. Nonlinear Sci. Numer. Simul. 19 (9), 2951 2957.
Angulo, J.M., Ruiz-Medina, M.D., Anh, V.V., Grecksch, W., 2000. Robust adaptive synchroni-
zation of different uncertain chaotic systems subject to input nonlinearity. Commun.
Nonlinear Sci. Numer. Simul. 15 (2), 430 441.
Asheghan, M.M., Beheshti, M.T.H., Tavazoei, M.S., 2011. Robust synchronization of perturbed
Chen’s fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16 (2),
1044 1051.
Astakhov, V.V., Anishchenko, V.S., Kapitaniak, T., Shabunin, A.V., 1997. Synchronization of chaotic
oscillators by periodic parametric perturbations. Phys. D Nonlinear Phenom. 109 (1), 11 16.
Atangana, A., Bildik, N., 2013. The use of fractional order derivative to predict the groundwater
flow. Math. Probl. Eng. 2013. Available from: https://doi.org/10.1155/2013/543026, Article
ID 543026, 9 pages.
Atangana, A., Vermeulen, P.D., 2014. Analytical solutions of a space-time fractional derivative
of groundwater flow equation. Abstr. Appl. Anal. 2014. Available from: https://doi.org/
10.1155/2014/381753, Article ID 381753, 11 pages.
Azar, A.T., Vaidyanathan, S., 2015a. Handbook of Research on Advanced Intelligent Control
Engineering and Automation. Advances in Computational Intelligence and Robotics (ACIR)
Book Series. IGI Global, USA, ISBN 9781466672482.
Azar, A.T., Vaidyanathan, S., 2015b. Computational Intelligence Applications in Modeling and
Control. Studies in Computational Intelligence, Vol. 575. Springer-Verlag, Germany, ISBN
978-3-319-11016-5.
Azar, A.T., Vaidyanathan, S., 2015c. Chaos Modeling and Control Systems Design, Studies in
Computational Intelligence, Vol. 581. Springer-Verlag, Germany, ISBN 978-3-319-13131-3.
Azar, A.T., Vaidyanathan, S., 2016. Advances in Chaos Theory and Intelligent Control. Studies in
Fuzziness and Soft Computing, Vol. 337. Springer-Verlag, Germany, ISBN 978-3-319-30338-3.