Abstract
Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same system’s parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.
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Grzesikiewicz, W., Wakulicz, A., Zbiciak, A.: Non-linear problems of fractional calculus in modeling of mechanical systems. Int. J. Mech. Sci. 70, 90–98 (2013)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Diethelm, K.: The Analysis of Fractional Differential Equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls, Fundamentals and Applications. Springer, London (2010)
Petras, I.: Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation. Higher Education Press and Springer, Beijing and Berlin (2011)
Ghasemi, S., Tabesh, A., Askari-Marnani, J.: Application of fractional calculus theory to robust controller design for wind turbine generators. IEEE Trans. Energy Convers. 29, 780–787 (2014)
Bhalekar, S.: Synchronization of incommensurate non-identical fractional order chaotic systems using active control. Eur. Phys. J. Spec. Top. 223, 1495–1508 (2014)
Sasso, A., Palmieri, G., Amodio, D.: Application of fractional derivative models in linear viscoelastic problems. Mech. Time Depend. Mater. 15, 367–387 (2011)
Guyomar, D., Ducharne, B., Sebald, G., Audiger, D.: Fractional derivative operators for modeling the dynamic polarization behavior as a function of frequency and electric field amplitude. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 437–443 (2009)
Zhang, R., Yang, S.: Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn. 66, 831–837 (2011)
Rakkiyappan, R., Velmurugan, G., Cao, J.: Stability analysis of memristor-based fractional-order neuronal networks with different memductance function. Cogn. Neurodyn. 9, 145–177 (2015)
Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)
Lu, J.G., Chen, G.: A note on the fractional order Chen system. Chaos Soliton Fractals 27, 685–688 (2006)
Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 42, 485–490 (1995)
Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A 341, 55–61 (2004)
Ahmad, W.M., Sprott, J.C.: Chaos in fractional order autonomous nonlinear systems. Chaos Soliton Fractals 16, 339–351 (2003)
Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoe analysis and circuit realization for a fractional-order Lü system. Nonlinear Dyn. 74, 203–212 (2013)
HosseinNia, S.H., Magin, R.L., Vinagre, B.M.: Chaos in fractional and integer order NSG systems. Signal Process. 107, 302–311 (2015)
Ma, T., Zhang, J.: Hybrid synchronization of coupled fractional-order complex networks. Neurocomputing 157, 166–172 (2015)
Kiani-B, A., Fallahi, K., Pariz, N., Leung, H.: A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Commun. Nonlinear Sci. 14, 863–879 (2009)
Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB). Nonlinear Dyn. 80, 1883–1897 (2015)
Xu, Y., Wang, H., Li, Y., Pei, B.: Image encryption based on synchronization of fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19, 3735–3744 (2014)
Muthukumar, P., Balasubramaniam, P.: Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dyn. 74, 1169–1181 (2013)
Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES). Nonlinear Dyn. 77, 1547–1559 (2014)
Campos-Canton, E., Barajas-Ramirez, J.G., Solis-Perales, G., Femat, R.: Multiscroll attractors by switching systems. Chaos 20, 013116/6 (2010)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
Yang, X.S., Yu, Y.G., Zhang, S.C.: A new proof for existence of horseshoe in the Rössler system. Chaos Soliton Fractals 18, 223–227 (2003)
Yang, X.S., Tang, Y.: Horseshoes in piecewise continuous maps. Chaos Soliton Fractals 19, 841–845 (2004)
Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Chaotic Characteristics analysis and circuit implementation for a fractional-order system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 61, 845–853 (2014)
Yang, X.S.: Topological horseshoes and computer assisted verification of chaotic dynamics. Int. J. Bifurc. Chaos 19, 1127–1145 (2009)
Wu, W.J., Chen, Z.Q., Yuan, Z.Z.: A computer-assisted proof for the existence of horseshoe in a novel chaotic system. Chaos Soliton Fractals 41, 2756–2761 (2009)
Diethelm, K., Neville, F., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)
Munoz-Pacheco, J.M., Zambrano-Serrano, E., Felix-Beltran, O.G., Gomez-Pavon, L.C., Luis-Ramos, A.: Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems. Nonlinear Dyn. 70, 1633–1643 (2012)
Munoz-Pacheco, J.M., Tlelo-Cuautle, E.: Simulation of Chua’s circuit by automatic control of step-size. Appl. Math. Comput. 190, 1526–1533 (2007)
Mohammad, S.T., Mohammad, H.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008)
Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. Chaos 9, 413 (1999)
Krishna, B.T., Reddy, K.V.V.S.: Active and passive realization of fractance device of order 1/2. Active and Passive Electronic Components 2008, Article ID 369421 (2008)
Acknowledgments
E. Zambrano-Serrano is a doctoral fellow of CONACYT (Mexico) in the Graduate Program on Control and Dynamical Systems at DMAp-IPICYT. E. Campos-Cantón acknowledges CONACYT for the financial support through Project No. 181002. J.M. Muñoz-Pacheco thanks VIEP-BUAP (No. MUPJ-ING15-G) to partially support this research.
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Zambrano-Serrano, E., Campos-Cantón, E. & Muñoz-Pacheco, J.M. Strange attractors generated by a fractional order switching system and its topological horseshoe. Nonlinear Dyn 83, 1629–1641 (2016). https://doi.org/10.1007/s11071-015-2436-z
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DOI: https://doi.org/10.1007/s11071-015-2436-z