Modified projective synchronization of fractional-order hyperchaotic memristor-based Chua’s circuit
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Abstract
This paper investigates the modified projective synchronization (MPS) between two hyperchaotic memristor-based Chua circuits modeled by two nonlinear integer-order and fractional-order systems. First, a hyperchaotic memristor-based Chua circuit is suggested, and its dynamics are explored using different tools, including stability theory, phase portraits, Lyapunov exponents, and bifurcation diagrams. Another interesting property of this circuit was the coexistence of attractors and the appearance of mixed-mode oscillations. It has been shown that one can achieve MPS with integer-order and incommensurate fractional-order memristor-based Chua circuits. Finally, examples of numerical simulation are presented, showing that the theoretical results are in good agreement with the numerical ones.
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References
A. Boulkroune and M. Msaad, On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign, Fuzzy Sets. Syst., 201 (2012), 71-85.
A. Boulkroune, S. Hamel, F. Zouari, and A. Leas, Output-Feedback Controller Based Projective Lag Synchronization of Uncertain Chaotic Systems in the Presence of Input Nonlinearities, Mathematical Problems in Engineering, 81 (2017), 1-12.
A. Chen, J. Lu, Jinhu Lü and S. Yu, Generating hyperchaotic Lü attractor via state feedback control, Physica A, 364 (2006), 103-110.
M. M. Al-Sawalha, and A. Al-Sawalha, Anti-synchronization of fractional-order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control, Open Phys., 14(1) (2016), 304-313.
A. Soukkou and S. Leulmi, Controlling and Synchronizing of Fractional-Order Chaotic Systems via Simple and Optimal Fractional-Order Feedback Controller, International Journal of Intelligent Systems Technologies and Applications., 8(6) (2016), 56-69.
A. Soukkou, A. Boukabou and S. Leulmi, Prediction-based feedback control and synchronization algorithm of fractional-order chaotic systems, Nonlinear Dyn. 58(4) (2016), 2183-2206.
A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determning Lyapunov Exponents from a Time Series, Physica D., 16 (1985), 285-317.
B. C. Bao, J. P. Xu and Z. Liu, Initial state dependent dynamical behaviors in memristor based chaotic circuit, Chinese Physics Letters, 27(7) (2010), 070504.
B. Muthuswamy and P. P. Kokate, Memristor based chaotic circuits, IETE Technical Review. 26(6) (2009), 415-426.
B. Wang, J. Jian and H. Yu, Adaptive synchronization of fractional-order memristor-based Chuas system, Systems Science, (2014), 291-296.
D. Cafagna and G. Grassi, On the simplest fractional-order memristor-based chaotic system, Nonlinear Dyn., 70(2)(2012), 1185-1197.
G. Ye and J. Zhou, A block chaotic image encryption scheme based on self-adaptive modelling, Applied Soft Computing, 22(2014), 351-357.
I. Petráš, Fractional-order memristor-based Chua’s circuit, IEEE Trans. Circuits Syst. II., 57(12) (2010), 975-979.
J. Borghetti, G. S. Snider , P. J. Kuekes, J. J. Yang, D. R. Stewart and R. S. Williams, Memristive switches enable tateful logic operations via material implication, 64(4) (2010), 873-876.
J. Wang and Z. Chen, A novel hyperchaotic system and its complex dynamics, Int. J. Bifurcation Chaos, 18(11) (2008), 3309-3324.
K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms., 36 (2004), 31-52.
K. Murali and M. Lakshmanan, Secure communication using a compound signal from generalized chaotic systems, Phys. Lett. A., 241(6) (1998), 303-310.
L. Jian , L. Shutang and Y. Chunhua, Modified generalized projective synchronization of fractional-order chaotic Lü systems, Adv. Differ. Equations, (2013), 2013-374.
L. Kocarev and U. Parlitz, General approach for chaotic synchronization with application to communication, Phys. Rev. Lett., 74 (1995), 5028-5031.
L. O. Chua, Memristor, the missing circuit element, IEEE Trans. circuit. Theory., 18 (1971), 507-519.
L. Teng, H. C. Iu Herbert , X. Y. Wang and X. K. Wang, Chaotic behavior in fractionalorder memristor-based simplest chaotic circuit using fourth degree polynomial, Nonlinear Dyn., 77(1-2) (2014), 231-241.
M-S. Abdelouahab, N-E. Hamri and J. Wang, Hopf bifurcation and chaos in fractional-order modified hybrid optical system, Nonlinear Dyn., 69(1) (2012), 275–284.
M-S. Abdelouahab and R. Lozi, Hopf bifurcation and chaos in simplest fractional-order memristors-based electrical circuit, Indian Journal of Industrial and Applied Mathematics, 6(2)(2015), 105-119.
M-S. Abdelouahab, R. Lozi and L. O. Chua, Memfractance: a mathematical paradigm for circuit elements with memory, Int. J. Bifurcation Chaos., 24(9) (2014), 1430023 (29 pages).
M. Bharathwaj, Khalil, Implementing memristor based chaotic circuit, Int. J. Bifurcation Chaos, 20(5) (2010), 1335-1350.
M. Itoh and L. O. Chua, Memristor Oscillators, Int. J. Bifurcation Chaos, 18(11) (2008), 3183-3206.
O. E. Rossler, An equation for hyperchaos, Phys. Lett. A., 71(2-3) (1979), 155-177.
Q. Li, S. Hu, S. Tang and G. Zeng, Hyperchaos and horseshoe in a 4 − D memristive system with a line of equilibria and its implementation, Int. J. Circuit Theory Appl., 42(11) (2014), 1172-1188.
R. Suresh, and V. Sundarapandian, Hybrid synchronization of n–scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback. Arch. Control Sci., 3 (2012), 343-365.
Sh. Wang, Xi. Wang and Y. Zhou, A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization, Entropy.,17(11) (2015), 7628-7644.
S. Kaouache and M-S. Abdelouahab, Modified Projective Synchronization between Integer-Order and fractional-order Hyperchaotic Systems, Journal of Adv Research in Dynamical and Control Systems, 10(5) (2018), 96-104.
S. Rasappan, Y. Li , X. Huang, Y. Song and J. Lin, A new fourth-order memristive chaotic system and its generation, Int. J. Bifurcation Chaos, 25(11) (2015), 1550151.
S. Shin, K. Kim and S. M. Kang , Memristor applications for programmable analog ICs, IEEE Transactions in Nanotechnology, 10(2) (2011), 266-274.
S. Vaidyanathan, Ch. K. Volos and V.-T. Pham, Analysis, Control, Synchronization and SPICE Implementation of a Novel 4-D Hyperchaotic Rikitake Dynamo System without Equilibrium, Journal of Engineering Science and Technology Review, 8 (2)(2015), 232-244.
S. Vaidyanathan, Ch. K. Volos and V. -T. Pham, Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation, Journal of Engineering Science and Technology Review, 8(2) (2015), 174–184.
T. I. Chien and T. L. Liao, Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization, Chaos, Solitons Fractals, 24 (2005), 241-245.
V. T. Pham, Ch. Volos and L. V. Gambuzza, A Memristive Hyperchaotic System without Equilibrium, The Scientific World Journal, 368986 (2014), 1-9.
W. Zhen, H. Xia and S. Hao, Control of an uncertain fractional-order economic system via adaptive sliding mode, Neurocomputing, 83 (2012), 83-88.
X. Huang, J. Jia, Y. Li and Z. Wang, Complex Nonlinear Dynamics in fractional and integer order memristor-based systems, Neurocomputing., 218(19) (2016), 296-306.
H. Xi, Y. Li, and X. Huang. Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems, Entropy, 16(12) (2014), 6240-6253.
X. J. Wu and Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dyn., 57(1-2) (2009), 25-35.
Y. V. Pershin and M. D. Ventra , Experimental demonstration of associative memory with memristive neural networks, Neural Networks, 23(7) (2010), 881-886.
Y. Yu and H. Li, The synchronization of fractional-order Rossler hyperchaotic systems, Physica A, 387 (5-6)(2008), 1393-1403.
Z. Elhadj, Dynamical Analysis of a 3 − D Chaotic System with only Two Quadratic Nonlinearities, J. Syst. Sci. Complex., 21(1) (2008), 67-75.
Z. Hrubš and T. Gotthans, Analysis and synthesis of chaotic circuits using memristor properies, Journal of electrical engineering, 65(3) (2014), 129-136.