Research Papers

Observe-Based Projective Synchronization of Chaotic Complex Modified Van Der Pol-Duffing Oscillator With Application to Secure Communication

[+] Author and Article Information
Ping Liu

College of Mechanical
and Electronic Engineering,
Shandong Key Laboratory
of Gardening Machinery and Equipment,
Shandong Agricultural University,
Taian 271018, China
College of Engineering,
Peking University,
Beijing 100871, China

Hongjun Song

College of Mechanical
and Electronic Engineering,
Shandong Agricultural University,
Taian 271018, China

Xiang Li

State Key Laboratory of Crop Biology,
College of Life Sciences,
Shandong Agricultural University,
Taian 271018, China
e-mail: lixiang@sdau.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 21, 2014; final manuscript received January 31, 2015; published online April 16, 2015. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 10(5), 051015 (Sep 01, 2015) (7 pages) Paper No: CND-14-1218; doi: 10.1115/1.4029715 History: Received September 21, 2014; Revised January 31, 2015; Online April 16, 2015

This paper addresses the projective synchronization (PS) of the complex modified Van der Pol-Duffing (MVDPD for short) chaotic oscillator by using the nonlinear observer control and also discusses its applications to secure communication in theory. First, we construct the complex MVDPD oscillator and analysis its chaotic behavior. Moreover, an observer design method is applied to achieve PS of two identical MVDPD chaotic oscillators with complex offset terms, which are synchronized to the desired scale factor. The unpredictability of the scaling factor could further enhance the security of the communication. Finally, numerical simulations are given to demonstrate the effectiveness and feasibility of the proposed synchronization approach and also verify the success application to the proposed scheme’s in the secure communication.

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Liao, T. L., and Huang, N. S., 1999, “An Observer Based Approach for Chaotic Synchronization and Secure Communication,” IEEE Trans. Circuits Syst. I, 46(9), pp. 1144–1149. [CrossRef]
Luo, A. C. J., 2009, “A Theory for Synchronization of Dynamical Systems,” Commun. Nonlinear Sci. Numer. Simul., 14(5), pp. 1901–1951. [CrossRef]
Parlitz, U., Kocarev, L., Stojanovski, T., and Preckel, H., 1996, “Encoding Messages Using Chaotic Synchronization,” Phys. Rev. E, 53(5), pp. 4351–4361. [CrossRef]
Chen, G., and Dong, X., 1998, From Chaos to Order: Methodologies, Perspectives, and Applications, World Scientific, Singapore. [CrossRef]
Alvarez, G., Li, S., Montoya, F., Pastor, G., and Romera, M., 2004, “Breaking a Secure Communication Scheme Based on the Phase Synchronization of Chaotic Systems,” Chaos, 14(2), pp. 274–278. [CrossRef] [PubMed]
Mahmoud, G., Kashif, M., and Farghaly, A., 2008, “Chaotic and Hyperchaotic Attractors of a Complex Nonlinear System,” J. Phys. A, 41(5), p. 055104. [CrossRef]
Mahmoud, G., and Aly, S. A., 2000, “Periodic Attractors of Complex Damped Non-Linear Systems,” Int. J. Nonlinear Mech., 35(2), pp. 309–323. [CrossRef]
Mahmoud, G. M., and Mahmoud, E. E., 2010, “Complete Synchronization of Chaotic Complex Nonlinear Systems With Uncertain Parameters,” Nonlinear Dyn., 62(4), pp. 875–882. [CrossRef]
Abarbanel, H. D. I., Rulkov, N. F., and Sushchik, M. M., 1996, “Generalized Synchronization of Chaos: The Auxiliary System Approach,” Phys. Rev. E, 53(5), pp. 4528–4535. [CrossRef]
Mainieri, R., and Rehacek, J., 1999, “Projective Synchronization in Three-Dimensional Chaotic Systems,” Phys. Rev. Lett., 82(15), pp. 3042–3045. [CrossRef]
Xu, D., 2001, “Control of Projective Synchronization in Chaotic Systems,” Phys. Rev. E, 63(2), p. 027201. [CrossRef]
Xu, D., Li, Z., and Bishop, S., 2001, “Manipulating the Scaling Factor of Projective Synchronization in Three-Dimensional Chaotic Systems,” Chaos, 11(3), pp. 439–442. [CrossRef] [PubMed]
Li, G. H., 2006, “Generalized Projective Synchronization of Two Chaotic Systems by Using Active Control,” Chaos Solitons Fract., 30(1), pp. 77–82. [CrossRef]
Shi, X., and Wang, Z., 2009, “Projective Synchronization of Chaotic Systems With Different Dimensions Via Backstepping Design,” Int. J. Nonlinear Sci., 7, pp. 301–306.
Ghosh, D., and Bhattacharya, S., 2010, “Projective Synchronization of New Hyperchaotic System With Fully Unknown Parameters,” Nonlinear Dyn., 61(1–2), pp. 11–21, 2010. [CrossRef]
Hua, M., Yang, Y., Xua, Z., and Guo, L., 2008, “Hybrid Projective Synchronization in a Chaotic Complex Nonlinear System,” Math. Comput. Simul., 79(3), pp. 449–457. [CrossRef]
Callier, F. M., and Desoer, C. A., 1991, Linear System Theory, Springer-Verlag, New York. [CrossRef]
Vidyasagar, M., 1993, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Darouach, M., and Boutayeb, M., 1995, “Design of Observers for Descriptor Systems,” IEEE Trans. Autom. Control, 40(7), pp. 1323–1327. [CrossRef]
Nijmeijer, H., and Mareels, I., 1997, “An Observer Looks at Synchronization,” IEEE Trans. Circuits Syst. I, 44(10), pp. 882–890. [CrossRef]
Grassi, G., and Mascolo, S., 1997, “Nonlinear Observer Design to Synchronize Hyperchaotic Systems Via a Scalar Signal,” IEEE Trans. Circuits Syst. I, 44(10), pp. 1011–1014. [CrossRef]
Liu, B., and Peng, J., 2004, Nonlinear Dynamics, Higher Education Press, Beijing, China.
Qiu, S., and Filanovsky, I. M., 1987, “Periodic Solution of Van Der Pol Equation With Moderate Values of Damping Coefficient,” IEEE Trans. Circuits Syst. I, 34(8), pp. 913–918. [CrossRef]
Sira-Ramirez, H., 2002, “Harmonic Response of Variable-Structure-Controlled Van der Pol Oscillators,” IEEE Trans. Circuits Syst. I, 34(1), pp. 103–106. [CrossRef]
Li, X., Ji, J. C., Hansen, C. H., and Tan, C., 2006, “The Response of a Duffing-Van Der Pol Oscillator Under Delayed Feedback Control,” J. Sound Vib., 291(3–5), pp. 644–655. [CrossRef]
Mahmoud, G. M., and Farghaly, A. A., 2004, “Chaos Control of Chaotic Limit Cycles of Real and Complex Van Der Pol Oscillators,” Chaos Solitons Fract., 21(4), pp. 915–924. [CrossRef]
Mahmoud, G. M., Mahmoud, E. E., Farghaly, A. A., and Aly, S. A., 2009, “Chaotic Synchronization of Two Complex Nonlinear Oscillators,” Chaos Solitons Fract., 42(5), pp. 2858–2864. [CrossRef]
Xu, Y., Xu, W., and Mahmoud, G. M., 2004, “On a Complex Beam–Beam Interaction Model With Random Forcing,” Physica A, 336(3–4), pp. 347–360. [CrossRef]
Liao, T., and Tsai, S., 2000, “Adaptive Synchronization of Chaotic Systems and Its Application to Secure Communications,” Chaos Solitons Fract., 11(9), pp. 1387–1396. [CrossRef]


Grahic Jump Location
Fig. 1

Chaotic attractors of the MVDPD oscillator as m=100,α=0.3,β=300,γ=0.2, and μ=0.2+0.05 j in different phase planes and projections. (a) The chaotic attractor in (z1r,z2r) projection. (b) The chaotic attractor in (z1i,z3i) projection. (c) The chaotic attractor in (z1r,z2r,z3r) space. (d) The chaotic attractor in (z1i,z2r,z3i) space.

Grahic Jump Location
Fig. 2

The states PS between systems (7) and (12). (a) The PS between the states e1 and e∧1. (b) The PS between the states e2 and e∧2. (c) The PS between the states e3 and e∧3.

Grahic Jump Location
Fig. 4

The secure communication via PS based on an observer

Grahic Jump Location
Fig. 5

Output signal of the transmitter for the parameters of Fig. 2

Grahic Jump Location
Fig. 6

The secure communication simulation results. (a) The transmitted signal. (b) Error evolution between recovery signal message and transmitted signal (the parameters are the same with Fig. 2).

Grahic Jump Location
Fig. 3

The error dynamics of the states from system (15). (a) The errors of the state e1. (b) The errors of the state e2. (c) The errors of the state e3.




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