Abstract
Synchronization of N-coupled fractional-order (FO) chaotic oscillators arranged in regular and irregular topologies is numerically studied. Synchronization is achieved based on the coupling matrix from the complex systems theory. In particular, we consider complex dynamical networks composed by Lorenz, Volta, Duffing and Financial FO chaotic oscillators, where the interaction of the nodes is defined by coupling only one state of each FO oscillator.
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A. B. Malinowska, T. Odzijewicz, and D. F. Torres, Advanced Methods in the Fractional Calculus of Variations, pp. 23–30, Springer International Publishing, 2015.
J. A. Machado and A. M. Lopes, “Analysis of natural and artificial phenomena using signal processing and fractional calculus,” Fractional Calculus and Applied Analysis, vol. 18, no. 2, pp. 459–478, 2015. [click]
V. Feliu-Batlle, R. Rivas-Perez, & F. J. Castillo-García, “Simple fractional order controller combined with a Smith predictor for temperature control in a steel slab reheating furnace,” International Journal of Control, Automation and Systems, vol. 11, no 3, p. 533–544, 2013. [click]
H. Bao and J. Cao, “Projective synchronization of fractional-order memristor-based neural networks,” Neural Networks, vol. 63, pp. 1–9, 2015. [click]
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, 1998.
T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer Science & Business Media, 2011.
E. Estrada, “Introduction to complex networks: structure and dynamics,”Evolutionary Equations with Applications in Natural Sciences, pp. 93–131, Springer International Publishing, 2015. [click]
A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou. “Synchronization in complex networks,” Physics Reports, vol. 469, no 3, pp. 93–153, 2008. [click]
X. F. Wang, “Complex networks: Topology, dynamic and synchronization,” International Journal of Bifurcation and Chaos, vol. 12, no. 5, pp. 885–916, 2002.
X. F. Wang and G. Chen, “Synchronization in small-world dynamical networks,” International Journal of Bifurcation and Chaos, vol. 12, no. 1, pp. 187–192, 2002. [click]
A. G. Soriano Sanchez, C. Posadas-Castillo, and M. A. Platas-Garza, “Synchronization of Generalized Chua’s Chaotic Oscillators in Small-world Topology,” Journal of Engineering Science and Technology Review, vol. 8, no. 2, pp. 185–191, 2015.
L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–825, 1990. [click]
H. Serrano-Guerrero, C. Cruz-Hernández, R.M. López-Gutiérrez, C. Posadas-Castillo, and E. Inzunza-González, “Chaotic synchronization in star coupled networks of three-dimensional cellular neural networks and its application in communications,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 8, pp. 571–580, 2010.
Y. Aguilar-Bustos, C. Cruz-Hernández, R. M. López-Gutiérrez, and C. Posadas-Castillo, “Synchronization of different hyperchaotic maps for encryption,” Nonlinear Dynamics and Systems Theory, vol. 8, no, 3, pp 221–236, 2008.
A. Kiani-B, K. Fallahi, N. Pariz, and H. Leung, “A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 863–879, 2009. [click]
J. Cao and Y. Wan, “Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays,” Neural Networks, vol. 53, pp. 165–172, 2014. [click]
J. Cao, A. Alofi, A. Al-Mazrooei, and A. Elaiw, “Synchronization of switched interval networks and applications to chaotic neural networks,” Abstract and Applied Analysis, vol. 2013, doi:10.1155/2013/940573, 2013. [click]
C. Zheng and J. Cao, “Robust synchronization of dynamical network with impulsive disturbances and uncertain parameters,” International Journal of Control, Automation and Systems, vol. 11, no 4, pp. 657–665, 2013. [click]
J. Cao, D. W. C. Ho, and Y. Yang, “Projective synchronization of a class of delayed chaotic systems via impulsive control,” Physics Letters A, vol. 373, no 35, pp. 3128–3133, 2009. [click]
Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,” International Journal of Bifurcation and Chaos, vol. 20, no. 1, pp. 81–97, 2010. [click]
G. Chen, “Pinning control and synchronization on complex dynamical networks,” International Journal of Control, Automation and Systems, vol. 12, no. 2, pp. 221–230, 2014. [click]
M. Liu, H. Chen, S. Zhang, and W. Sheng, “H ∞ synchronization of two different discrete-time chaotic systems via a unified model,” International Journal of Control, Automation and Systems, vol. 13, no. 1, pp. 212–221, 2015. [click]
E. Naseri, A. Ranjbar, and S. H. HosseinNia, “Backstepping control of fractional-order chen system,” Proc. of ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, pp. 1181–1186. January, 2009.
S. Y. Angulo-Guzman, C. Posadas-Castillo, D. A. Diaz-Romero, R. M. Lopez-Gutierrez, and C. Cruz-Hernandez, “Chaotic synchronization of regular complex networks with fractional-order oscillators,” Proc. of Control & Automation (MED) 20th Mediterranean Conference on, July, 2012.
I. Petras, Fractional Order Nonlinear Systems, Springer, 2011.
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Recommended by Associate Editor Choon Ki Ahn under the direction of Editor Ju Hyun Park. This work was supported by CONACYT, México under Research Grant No. 166654; PAICYT, México under Research Grant IT956-11; PROMEP, México under Research Grant No. 103.5/11/4330; and by “Facultad de Ingeniería Mecánica y Eléctrica”.
Sara Angulo-Guzman was born in Cd. Obregón, Sonora, México in 1987. She received the B.Sc. degree from the Technological Institute of Sonora (ITSON) in 2009 and the M.Sc. degree electrical engineering from Autonomous University of Nuevo Leon (UANL) in 2012. She is currently a professor at the Department of Electrical and Electronic Engineering at the Technological Institute of Sonora. Her research interests include synchronization and control of complex dynamical systems.
Cornelio Posadas-Castillo received the Engineer Degree in Control and Computation from the Autonomous University of Nuevo León, in 1997, Master in Science Degree in Electronics and Telecommunications, from CICESE in 2001, and Ph.D. degree in electrical from Baja California Autonomous University, in 2008. Since 1997, he has been Associated Professor of the University Autonomous of Nuevo León, México. His research interests include Chaos Synchronization, control of complex systems, nonlinear systems analysis, and private communications.
Miguel Angel Platas-Garza received his Ph.D. degree in Electrical Engineering from the Universidad Autónoma de Nuevo León (UANL), México in 2011. He holds currently an Associate Professor position at the UANL.
David Alejandro Diaz-Romero received his BSc (Eng.) and MSc. degrees from Universidad Autónoma de Nuevo León, Mexico and his PhD degree in automatic control theory from The University of Sheffield, U.K. He has worked in industry and academics. He currently holds a full time researcher position at Universidad Autónoma de Nuevo León, Mexico.
Didier Lopez-Mancilla received his Ph.D. degree in Electronics and Telecommunications from Scientific Research and Advances Studies of Ensenada, CICESE, México, in 2005. He has been working for University of Guadalajara as a researcher since 2006. He is a Professor of Control Theory for Mechatronics Engineering and he is currently the leader of the research group Applications in Optics and Electronics.
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Angulo-Guzman, S., Posadas-Castillo, C., Platas-Garza, M.A. et al. Chaotic synchronization of regular and irregular complex networks with fractional order oscillators. Int. J. Control Autom. Syst. 14, 1114–1123 (2016). https://doi.org/10.1007/s12555-015-0168-y
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DOI: https://doi.org/10.1007/s12555-015-0168-y