# American Institute of Mathematical Sciences

March  2020, 25(3): 1001-1013. doi: 10.3934/dcdsb.2019205

## Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form

 1 Centre for Non-Linear Dynamics, Defense University, Ethiopia 2 Institute of Energy, Mekelle University, Mekelle, Ethiopia 3 Department of Electronic and Automation, Vocational School of Hacibektas, Nevsehir Haci Bektas Veli University, 50800, Hacibektas, Nevsehir, Turkey 4 Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, 54050 Serdivan, Sakarya, Turkey 5 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: Akif Akgul

Received  March 2018 Revised  March 2019 Published  March 2020 Early access  September 2019

There are many works on self-excited and hidden attractors. However the relationship between them is less investigated. In this study we present a system which can have both hidden self-excited attractors. Dynamical properties of the chaotic system are studied using the equilibrium points and Eigenvalues analysis, Lyapunov exponents and bifurcation plots. Since fractional order models are more interesting in engineering applications, the fractional order version of the proposed system is derived using Adomian decomposition method. Bifurcation and stability analysis of the fractional order model shows the existence of chaotic oscillations. To demonstrate the engineering importance of the fractional order model, we have designed a digital communication system with SCSK (Symmetric Chaos Shift Keying) modulation method separately for both self-excited attractor and hidden attractor, the bit error rate performance was compared.

Citation: Karthikeyan Rajagopal, Serdar Cicek, Akif Akgul, Sajad Jafari, Anitha Karthikeyan. Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1001-1013. doi: 10.3934/dcdsb.2019205
##### References:
 [1] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling, 13 (1990), 17-43.  doi: 10.1016/0895-7177(90)90125-7. [2] M. Ahmad, U. Shamsi and I. R. Khan, An enhanced image encryption algorithm using fractional chaotic systems, Procedia Computer Science, 57 (2015), 852-859. [3] R. Caponetto and S. Fazzino, An application of adomian decomposition for analysis of fractional-order chaotic systems, International Journal of Bifurcation and Chaos, 23 (2013), 1350050, 7pp. doi: 10.1142/S0218127413500508. [4] A. Charef, H. H. Sun, Y. Y. Tsao and B. Onaral, Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, 37 (1992), 1465-1470.  doi: 10.1109/9.159595. [5] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Physics Reports, 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002. [6] S. He, K. Sun and H. Wang, Complexity analysis and dsp implementation of the fractional-order lorenz hyperchaotic system, Entropy, 17 (2015), 8299-8311.  doi: 10.3390/e17127882. [7] R. C. Hilborn et al., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, The Clarendon Press, Oxford University Press, New York, 1994. [8] M. A. Jafari, E. Mliki, A. Akgul, V.-T. Pham, S. T. Kingni, X. Wang and S. Jafari, Chameleon: The most hidden chaotic flow, Nonlinear Dynamics, 88 (2017), 2303-2317.  doi: 10.1007/s11071-017-3378-4. [9] S. T. Kingni, S. Jafari, H. Simo and P. Woafo, Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form, The European Physical Journal Plus, 129 (2014), 76. doi: 10.1140/epjp/i2014-14076-4. [10] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden chuas attractors, Physics Letters A, 375 (2011), 2230-2233.  doi: 10.1016/j.physleta.2011.04.037. [11] G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev, Hidden attractor and homoclinic orbit in lorenz-like system describing convective fluid motion in rotating cavity, Communications in Nonlinear Science and Numerical Simulation, 28 (2015), 166-174.  doi: 10.1016/j.cnsns.2015.04.007. [12] V.-T. Pham, S. Vaidyanathan, C. K. Volos, S. Jafari, N. V. Kuznetsov and T. M. Hoang, A novel memristive time–delay chaotic system without equilibrium points, The European Physical Journal Special Topics, 225 (2016), 127-136. [13] V.-T. Pham, S. Jafari, C. Volos, A. Giakoumis, S. Vaidyanathan and T. Kapitaniak, A chaotic system with equilibria located on the rounded square loop and its circuit implementation, IEEE Transactions on Circuits and Systems II: Express Briefs, 63 (2016), 878-882.  doi: 10.1109/TCSII.2016.2534698. [14] K. Rajagopal, A. Karthikeyan and P. Duraisamy, Hyperchaotic chameleon: Fractional order fpga implementation, Complexity, 2017 (2017), Art. ID 8979408, 16 pp. doi: 10.1155/2017/8979408. [15] D. Ruelle and F. Takens, On the nature of turbulence, Les Rencontres Physiciens-mathématiciens de Strasbourg-RCP25, 12 (1971), 1–44. [16] H. H. Sun, A. Abdelwahab and B. Onaral, Linear approximation of transfer function with a pole of fractional power, IEEE Transactions on Automatic Control, 29 (1984), 441-444.  doi: 10.1109/TAC.1984.1103551. [17] M. Sushchik, L. S. Tsimring and A. R. Volkovskii, Performance analysis of correlation-based communication schemes utilizing chaos, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47 (2000), 1684-1691.  doi: 10.1109/81.899920. [18] M. S. Tavazoei and M. Haeri, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems, IET Signal Processing, 1 (2007), 171-181.  doi: 10.1049/iet-spr:20070053. [19] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9. [20] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific, 2016.

show all references

##### References:
 [1] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling, 13 (1990), 17-43.  doi: 10.1016/0895-7177(90)90125-7. [2] M. Ahmad, U. Shamsi and I. R. Khan, An enhanced image encryption algorithm using fractional chaotic systems, Procedia Computer Science, 57 (2015), 852-859. [3] R. Caponetto and S. Fazzino, An application of adomian decomposition for analysis of fractional-order chaotic systems, International Journal of Bifurcation and Chaos, 23 (2013), 1350050, 7pp. doi: 10.1142/S0218127413500508. [4] A. Charef, H. H. Sun, Y. Y. Tsao and B. Onaral, Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, 37 (1992), 1465-1470.  doi: 10.1109/9.159595. [5] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Physics Reports, 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002. [6] S. He, K. Sun and H. Wang, Complexity analysis and dsp implementation of the fractional-order lorenz hyperchaotic system, Entropy, 17 (2015), 8299-8311.  doi: 10.3390/e17127882. [7] R. C. Hilborn et al., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, The Clarendon Press, Oxford University Press, New York, 1994. [8] M. A. Jafari, E. Mliki, A. Akgul, V.-T. Pham, S. T. Kingni, X. Wang and S. Jafari, Chameleon: The most hidden chaotic flow, Nonlinear Dynamics, 88 (2017), 2303-2317.  doi: 10.1007/s11071-017-3378-4. [9] S. T. Kingni, S. Jafari, H. Simo and P. Woafo, Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form, The European Physical Journal Plus, 129 (2014), 76. doi: 10.1140/epjp/i2014-14076-4. [10] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden chuas attractors, Physics Letters A, 375 (2011), 2230-2233.  doi: 10.1016/j.physleta.2011.04.037. [11] G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev, Hidden attractor and homoclinic orbit in lorenz-like system describing convective fluid motion in rotating cavity, Communications in Nonlinear Science and Numerical Simulation, 28 (2015), 166-174.  doi: 10.1016/j.cnsns.2015.04.007. [12] V.-T. Pham, S. Vaidyanathan, C. K. Volos, S. Jafari, N. V. Kuznetsov and T. M. Hoang, A novel memristive time–delay chaotic system without equilibrium points, The European Physical Journal Special Topics, 225 (2016), 127-136. [13] V.-T. Pham, S. Jafari, C. Volos, A. Giakoumis, S. Vaidyanathan and T. Kapitaniak, A chaotic system with equilibria located on the rounded square loop and its circuit implementation, IEEE Transactions on Circuits and Systems II: Express Briefs, 63 (2016), 878-882.  doi: 10.1109/TCSII.2016.2534698. [14] K. Rajagopal, A. Karthikeyan and P. Duraisamy, Hyperchaotic chameleon: Fractional order fpga implementation, Complexity, 2017 (2017), Art. ID 8979408, 16 pp. doi: 10.1155/2017/8979408. [15] D. Ruelle and F. Takens, On the nature of turbulence, Les Rencontres Physiciens-mathématiciens de Strasbourg-RCP25, 12 (1971), 1–44. [16] H. H. Sun, A. Abdelwahab and B. Onaral, Linear approximation of transfer function with a pole of fractional power, IEEE Transactions on Automatic Control, 29 (1984), 441-444.  doi: 10.1109/TAC.1984.1103551. [17] M. Sushchik, L. S. Tsimring and A. R. Volkovskii, Performance analysis of correlation-based communication schemes utilizing chaos, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47 (2000), 1684-1691.  doi: 10.1109/81.899920. [18] M. S. Tavazoei and M. Haeri, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems, IET Signal Processing, 1 (2007), 171-181.  doi: 10.1049/iet-spr:20070053. [19] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9. [20] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific, 2016.
Self-excited attractor
Hidden attractor
a):Bifurcation plot of the chaotic cuttlefish with parameter $b$; b): The corresponding finite time Lyapunov exponents; c): Cross section of basin of attraction of the system at $y = 1$ for $a = 0.8$; $b = 0$ d): Cross section of basin of attraction of the system at $y = 1$ for $a = 0.8$; $b = 0.2$ The notations $P, C, T, U$ in the plots c, d denotes the regions of initial conditions leading to periodic, chaotic, tori and unbounded oscillations respectively
2D phase portraits of the self-excited fractional order chaotic cuttlefish with the commensurate fractional order of $q = 0.998$ and $a = 1.493, b = 0.6, m = 4$
2D phase portraits of the hidden attractor fractional order chaotic cuttlefish with the commensurate fractional order of $q = 0.996$ and $a = 1.493, b = -0.8, m = 4$
Bifurcation of FOC cuttlefish (self-excited) system with order $q$
Bifurcation of FOC cuttlefish (hidden) system with order $q$
SCSK block diagrams of transmitter unit and receiver unit [17]
Matlab-Simulink® block diagram of the transmitter unit of the SCSK modulated communication system
Matlab-Simulink® block diagram of the receiver unit of the SCSK modulated communication system
Signals of the SCSK modulated communication system using self-excited attractor of fractional order chaotic cuttlefish system (a) transmitted (b) SCSK modulated (c) SCSK modulated with noise (d) retrieved
Signals of the SCSK modulated communication system using hidden attractor of fractional order chaotic cuttlefish system (a) transmitted (b) SCSK modulated (c) SCSK modulated with noise (d) retrieved
Comparison of BER performances of the SCSK modulated communication system
Chaotic cuttlefish
 Chaotic system System name Parameters Type of System $\begin{array}{l} \dot x = y&\\ \dot y = - 4x - 4yz&\\ \dot z = {x^2} + {y^2} + b{z^2} - a&\end{array}$ SA $a = 0.6 - 1.5$ Self-excited attractor $(b>0)$ HA $a = 0.6 - 1.5$ Hidden attractor $(b \le 0)$
 Chaotic system System name Parameters Type of System $\begin{array}{l} \dot x = y&\\ \dot y = - 4x - 4yz&\\ \dot z = {x^2} + {y^2} + b{z^2} - a&\end{array}$ SA $a = 0.6 - 1.5$ Self-excited attractor $(b>0)$ HA $a = 0.6 - 1.5$ Hidden attractor $(b \le 0)$
LEs and KY dimension of cuttlefish systems
 Attractor LEs KY dimension Self-excited ${{\rm{L}}_{\rm{1}}}{\rm{ = }}0.0144, {{\rm{L}}_2}{\rm{ = 0 }}{\rm{, }}{{\rm{L}}_3} = - 0.0421$ 2.342 Hidden ${{\rm{L}}_{\rm{1}}}{\rm{ = }}0.01865, {{\rm{L}}_2}{\rm{ = 0 }}{\rm{, }}{{\rm{L}}_3} = - 0.0372$ 2.501
 Attractor LEs KY dimension Self-excited ${{\rm{L}}_{\rm{1}}}{\rm{ = }}0.0144, {{\rm{L}}_2}{\rm{ = 0 }}{\rm{, }}{{\rm{L}}_3} = - 0.0421$ 2.342 Hidden ${{\rm{L}}_{\rm{1}}}{\rm{ = }}0.01865, {{\rm{L}}_2}{\rm{ = 0 }}{\rm{, }}{{\rm{L}}_3} = - 0.0372$ 2.501
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