Synchronization of a Fractional Order Chaotic System with Attractors Propagating on a Line (I)

Resumen: This paper addresses the problem of synchronization for a class of fractional order chaotic systems with a double-scroll attractor propagates on a line. Based on the stability theory of fractional order systems, some basic dynamical properties are studied, such as equilibrium points, Lyapunov exponents and strange attractors of the chaotic system. Using the active control approach the synchronization of incommensurate fractional order systems with different basins of attraction is achieved. The numerical results illustrate that fast synchronization can be achieved between fractional order chaotic systems.

¿Cómo citar?

E. Zambrano-Serrano, J.M. Muñoz-Pacheco, C. Posadas-Castillo, L.C. Gómez-Pavón & R.R. Rivera-Durón. Synchronization of a Fractional Order Chaotic System with Attractors Propagating on a Line (I). Memorias del Congreso Nacional de Control Automático, pp. 570-575, 2018.

Palabras clave

Chaotic behavior; Fractional order; Active control; Synchronization

• Abernethy, S. and Gooding, R. (2018). The importance of chaotic attractors in modelling tumour growth. Physica A: Statistical Mechanics and its Applications, 507, 268– 277.
• Bai, E. and Lonngren, K.E. (1997). Synchronization of two lorenz systems using active control. Chaos, Solitons Fractals, 8(1), 51–58.
• Cicek, S., Kocamaz, U., and Uyaroglu, Y. (2018). Secure communication with a chaotic system owning logic element. AEU – International Journal of Electronics and Communications, 88, 52–62.
• Diethelm, K. and Ford, N.J. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265, 229–248.
• Garrapa, R. (2018). Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics, 6(2), 1–16.
• Grigorenko, I. and Grigorenko, E. (2003). Chaotic dynamics of the fractional lorenz system. Phys. Rev. Lett, 91, 034101.
• Guo, Z., Si, G., Diao, L., Jia, L., and Zhang, Y. (2018). Generalized modeling of the fractional-order memcapacitor and its character analysis. Communications in Nonlinear Science and Numerical Simulation, 59, 177– 189.
• Hirsch, M. and Smale, S. (1965). Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York.
• Hsiao, F.H. (2018). Chaotic synchronization cryptosystems combined with rsa encryption algorithm. Fuzzy Sets and Systems, 342, 109–137.
• Ionescu, C. and Kelly, J.F. (2017). Fractional calculus for respiratory mechanics: Power law impedance, viscoelasticity, and tissue heterogeneity. Chaos, Solitons Fractals, 102, 433–440.
• Lan, R., J. He and, S.W., Gu, T., and Luo, X. (2018). Integrated chaotic systems for image encryption. Signal Processing, 147, 133–145.
• Li, C. and Chen, G. (2004). Chaos in the fractional order chen system and its control. Chaos, Solitons Fractals, 22(3), 549–554.
• Long, Y., Xu, B., Chen, D., and Ye, W. (2018). Dynamic characteristics for a hydro-turbine governing system with viscoelastic materials described by fractional calculus. Applied Mathematical Modelling, 58, 128–139.
• Munoz-Pacheco, J., Zambrano-Serrano, E., Volos, C., Tacha, O., Stouboulos, I., and Pham, V.T. (2018). A fractional order chaotic system with a 3d grid of variable attractors. Chaos, Solitons Fractals, 113, 69– 78.
• Pecora, L.M. and Carroll, T.L. (1990). Synchronization in chaotic systems,. Physical Review Letters, 64(8), 821– 824.
• Petras, I. (2011). Fractional-Order Nonlinear Systems. Springer International Publishing.
• Pham, V.T., Ouannas, A., Volos, C., and Kapitaniak, T. (2018). A simple fractional-order chaotic system without equilibrium and its synchronization. AEU – International Journal of Electronics and Communications, 86, 69–76.
• Rasmussen, R., Jensen, M., and Heltberg, M. (2017). Chaotic dynamics mediate brain state transitions, driven by changes in extracellular ion concentrations. Cell Systems, 5(6), 591–603.
• Saadia, A. and Rashdi, A. (2018). Incorporating fractional calculus in echo-cardiographic image denoising. Computers Electrical Engineering,, 67, 134–144.
• Shukla, M. and Sharma, B. (2017). Backstepping based stabilization and synchronization of a class of fractional order chaotic systems. Chaos, Solitons Fractals, 102, 274–284.
• Singh, A.K., Yadav, V.K., and Das, S. (2017). Synchronization between fractional order complex chaotic systems with uncertainty. Optik, 133, 98–107.
• Sun, H., Zhang, Y., Baleanu, D., Chen, W., and Chen, Y. (2018). A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 64, 213–231.
• Zambrano-Serrano, E., Campos-Canton, E., and Munoz Pacheco, J.M. (2016). Strange attractors generated by a fractional order switching system and its topological horseshoe. Nonlinear Dynamics, 83(3), 1629–1641.