Abstract
In this paper, we study the dynamics of the fractional-order chaotic system corresponding to the original Chua system with the same nonlinearity. We place bounds on the fractional order to guarantee a chaotic behavior. In addition, we propose a one-dimensional adaptive synchronization strategy, whereby we assume knowledge of one of the states and reconstruct the rest. The proposed synchronization scheme is put to the test in a secure communication scenario based on the antipodal chaos shift keying modulation scheme. Throughout the analysis and examples, numerical results are presented to affirm the validity of the findings.
References
[1] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci. 20(2) (1963), 130–141.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2Search in Google Scholar
[2] L. M. Pecora and T. L. Carrol, Synchronization in chaotic systems, Phys. Rev. A 64 (1990), 821–824.10.1016/B978-012396840-1/50040-0Search in Google Scholar
[3] L. O. Chua, M. Itoh, L. Kocarev and K. Eckert, Chaos synchronization in Chua’s circuit, J. Circuits Syst. Comput. 3(1) (1993), 93–108.10.1142/S0218126693000071Search in Google Scholar
[4] R. N. Madan, Chua’s circuit: a paradigm for chaos, World Sci. Series Nonlinear Sci. Series B: 1 (1993).10.1142/1997Search in Google Scholar
[5] R. Van Der Steen, Numerical and experimental analysis of multiple Chua circuits, MSc Thesis, Eindhoven University of Technology, 2006.Search in Google Scholar
[6] K. Murali and M. Lakshmanan, Synchronizing chaos in driven Chua’s circuit, Int. J. Bifurcation Chaos 3(4) (1993), 1057–1066.10.1142/S021812749300088XSearch in Google Scholar
[7] M. P. Kennedy, Bifurcation and chaos, in: W. K. Chen, editor, The circuits and filters handbook, IEEE Press, USA, 1995.Search in Google Scholar
[8] G. Chen and X. Dong, From chaos to order: methodologies, perspectives and applications, World Scientific Publishing Co. Pte. Ltd., Singapore, 1998.10.1142/3033Search in Google Scholar
[9] M. Feki, An adaptive chaos synchronization scheme applied to secure communication, Chaos, Solitons Fractals 18(1) (2003), 141–148.10.1016/S0960-0779(02)00585-4Search in Google Scholar
[10] L. Kocarev, K. S. Halle, K. Eckert, L. O. Chua and U. Parlitz, Applications of Chua’s Circuit, in: R.N. Madan editor, Chua’s Circuit: A paradigm for Chaos, World Scientific Publishing Co. Pte. Ltd., Singapore, 1993.Search in Google Scholar
[11] T. T. Hartley, C. F. Lorenzo and H. K. Qammer, Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst. I 42(8) (1995), 485–490.10.1109/81.404062Search in Google Scholar
[12] D. Cafagna and G. Grassi, Fractional order Chua’s circuit: time domain analysis, bifurcation, chaotic behavior, and test for chaos, Int. J. Bifurcation Chaos 18(3) (2008), 615–639.10.1142/S0218127408020550Search in Google Scholar
[13] Z. Odibat, N. Corson, M. A. Aziz–Alaoui and A. Alsaedi, Chaos in fractional order cubic Chua system and synchronization, Int. J. Bifurcation Chaos 27(10) (2017), 1750161.10.1142/S0218127417501619Search in Google Scholar
[14] S. Liu and F. Zhang, Complex function projective synchronization of complex chaotic system and its applications in secure communication, Nonlinear Dyn. 76(2) (2014), 1087–1097.10.1007/s11071-013-1192-1Search in Google Scholar
[15] F. Zhang and S. Liu, Self time-delay synchronization of time-delay coupled complex chaotic system and its applications to communication, Int. J. Modern Phys. C 25(3) (2014), 1350102.10.1142/S0129183113501027Search in Google Scholar
[16] F. Zhang, Complete synchronization of coupled multiple-time-delay complex chaotic system with applications to secure communication, Acta Physica Polonica B 46(8) (2015), 1473–1486.10.5506/APhysPolB.46.1473Search in Google Scholar
[17] F. Zhang, Lag synchronization of complex Lorenz system with applications to communication, Entropy 17(7) (2015), 4974–4985.10.3390/e17074974Search in Google Scholar
[18] A. Kiani, F. Kia Fallahi, N. Pariz and H. Leung, A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter, Commun. Nonlinear Sci. Numer. Simul. 14(3) (2009), 863–879.10.1016/j.cnsns.2007.11.011Search in Google Scholar
[19] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.Search in Google Scholar
[20] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, Robustness and signal recovery in a synchronized chaotic system, Int. J. Bifurcation Chaos 3(6) (1993), 1629–1638.10.1142/S021812749300129XSearch in Google Scholar
[21] H. Dedieu, M. P. Kennedy and M. Hasler, Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits, IEEE Trans. Circuits Syst. II: Analog Digital SP 40(10) (1993), 634–642.10.1109/82.246164Search in Google Scholar
[22] W. Hu, L. Wang and G. Kaddoum, Design and performance analysis of a differentially spatial modulated Chaos shift keying modulation system, IEEE Trans. Circuits Syst. II: Express Briefs 64(11) (2017), 1302–1306.10.1109/TCSII.2017.2697456Search in Google Scholar
[23] W. Xu, Y. Tan, F. C. M. Lau and G. Kolumban, Design and optimization of differential Chaos shift keying scheme with code index modulation, IEEE Trans. Commun. 66(5) (2018), 1970–1980.10.1109/TCOMM.2018.2805342Search in Google Scholar
[24] M. Engels, Wireless OFDM systems, Int. Series Eng. & Comp. Sci. (SECS) 692 (2002), Springer.10.1007/b117438Search in Google Scholar
[25] B. Chen, L. Zhang and H. Lu, High security differential Chaos-based modulation with channel scrambling for WDM-Aided VLC system, IEEE Photonics J. 8(5) (2016), 1–13.10.1109/JPHOT.2016.2607689Search in Google Scholar
[26] N. Jiang, C. Xue, C. Zhang and K. Qiu, Physical-enhanced secure communication based on wavelength division multiplexing chaos synchronization of multimode semiconductor lasers, IEEE/CIC Int. Conf. Comms. in China (ICCC) (2016), 1–5.10.1109/ICCChina.2016.7636797Search in Google Scholar
[27] D. He and H. Leung, Quasi-orthogonal chaotic CDMA multi-user detection using optimal chaos synchronization, IEEE Trans. Circuits and Systems II: Express Briefs 52(11) (2005), 739–743.10.1109/TCSII.2005.852538Search in Google Scholar
[28] W. Tam, F. Lao and C. Tse, Digital communications with chaos: multiple access techniques and performance, Elsevier Ltd., 2007.Search in Google Scholar
[29] T. Yang and L. Chua, Impulsive stabilization for control and synchronization of chaotic systems, theory and applications to secure communications, IEEE Trans. Circuits Syst. I 44(10) (1997), 976–988.10.1109/81.633887Search in Google Scholar
[30] G. Millerioux and C. Mira, Coding scheme based on chaos sychronization from noninvertible maps, Int. J. Bifurcation Chaos 8(10) (1998), 2019–2029.10.1142/S0218127498001674Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions
- Conservation Laws of Space-Time Fractional mZK Equation for Rossby Solitary Waves with Complete Coriolis Force
- Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation
- Multiple Solitary Wave Solutions for Nonhomogeneous Quasilinear Schrödinger Equations
- Fuzzy Logic Controller for Obstacle Avoidance of Mobile Robot
- Optimal Thrust Programming Along the Brachistochronic Trajectory with Non-linear Drag
- Numerical Treatment of the Fractional Modeling on Susceptible-Infected-Recovered Equations with a Constant Vaccination Rate by Using GEM
- The Fractional Chua Chaotic System: Dynamics, Synchronization, and Application to Secure Communications
- Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions
- Conservation Laws of Space-Time Fractional mZK Equation for Rossby Solitary Waves with Complete Coriolis Force
- Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation
- Multiple Solitary Wave Solutions for Nonhomogeneous Quasilinear Schrödinger Equations
- Fuzzy Logic Controller for Obstacle Avoidance of Mobile Robot
- Optimal Thrust Programming Along the Brachistochronic Trajectory with Non-linear Drag
- Numerical Treatment of the Fractional Modeling on Susceptible-Infected-Recovered Equations with a Constant Vaccination Rate by Using GEM
- The Fractional Chua Chaotic System: Dynamics, Synchronization, and Application to Secure Communications
- Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species