×

CAMO: self-excited and hidden chaotic flows. (English) Zbl 1439.34046

Summary: In this paper, we announce a novel 4D chaotic system which belongs to the self-excited attractor and hidden attractor family depending on the parameter values. Lyapunov exponents, bifurcation diagram and bicoherence plot of the CAMO (Camouflage) chaotic system are investigated. Also, fractional-order model of the proposed CAMO system (FOCAMO) is derived and analyzed. FOCAMO chaotic system is then implemented in Field Programmable Gate Array (FPGA) using Adomian decomposition method. Also, power efficiency analysis for various fractional-orders is investigated. The paper helps build a better understanding of chaotic systems with self-excited or hidden attractors.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A08 Fractional ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adomian, G. [1990] “ A review of the decomposition method and some recent results for nonlinear equations,” Math. Comput. Model.13, 17-43. · Zbl 0713.65051
[2] Aghababa, M. P. [2012] “ Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory,” J. Comput. Nonlin. Dyn.7, 021010.
[3] Baleanu, D., Diethelm, K., Scalas, E. & Trujillo, J. J. [2012] Fractional Calculus: Models and Numerical Methods, , Vol. 3 (World Scientific). · Zbl 1248.26011
[4] Boroujeni, E. A. & Momeni, H. R. [2012] “ Non-fragile nonlinear fractional-order observer design for a class of nonlinear fractional-order systems,” Sign. Process.92, 2365-2370.
[5] Brezetskyi, S., Dudkowski, D. & Kapitaniak, T. [2015] “ Rare and hidden attractors in van der Pol-Duffing oscillators,” Eur. Phys. J.224, 1459-1467.
[6] Cafagna, D. & Grassi, G. [2015] “ Fractional-order systems without equilibria: The first example of hyperchaos and its application to synchronization,” Chin. Phys. B24, 080502.
[7] Caponetto, R. & Fazzino, S. [2013] “ An application of adomian decomposition for analysis of fractional-order chaotic systems,” Int. J. Bifurcation and Chaos23, 1350050-1-7. · Zbl 1270.34010
[8] Charef, A., Sun, H., Tsao, Y. & Onaral, B. [1992] “ Fractal system as represented by singularity function,” IEEE Trans. Autom. Contr.37, 1465-1470. · Zbl 0825.58027
[9] Danca, M.-F., Tang, W. K. & Chen, G. [2016] “ Suppressing chaos in a simplest autonomous memristor-based circuit of fractional-order by periodic impulses,” Chaos Solit. Fract.84, 31-40. · Zbl 1355.94099
[10] Diethelm, K. [2010] The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer Science & Business Media). · Zbl 1215.34001
[11] Dong, E., Liang, Z., Du, S. & Chen, Z. [2016] “ Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement,” Nonlin. Dyn.83, 623-630.
[12] He, S., Sun, K. & Wang, H. [2015] “ Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system,” Entropy17, 8299-8311.
[13] Jafari, S., Sprott, J. C., Pham, V.-T., Golpayegani, S. M. R. H. & Jafari, A. H. [2014] “ A new cost function for parameter estimation of chaotic systems using return maps as fingerprints,” Int. J. Bifurcation and Chaos24, 1450134-1-18. · Zbl 1302.34027
[14] Jafari, S., Pham, V.-T. & Kapitaniak, T. [2016a] “ Multiscroll chaotic sea obtained from a simple 3D system without equilibrium,” Int. J. Bifurcation and Chaos26, 1650031-1-7. · Zbl 1334.34034
[15] Jafari, S., Sprott, J., Pham, V.-T., Volos, C. & Li, C. [2016b] “ Simple chaotic 3D flows with surfaces of equilibria,” Nonlin. Dyn.86, 1349-1358.
[16] Jafari, S., Sprott, J. C. & Molaie, M. [2016c] “ A simple chaotic flow with a plane of equilibria,” Int. J. Bifurcation and Chaos26, 1650098-1-6. · Zbl 1343.34037
[17] Kapitaniak, T. & Leonov, G. A. [2015] “ Multistability: Uncovering hidden attractors,” Eur. Phys. J.224, 1405-1408.
[18] Kingni, S., Jafari, S., Simo, H. & Woafo, P. [2014] “ Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form,” Eur. Phys. J.129, 76.
[19] Leonov, G., Kuznetsov, N. & Vagaitsev, V. [2011] “ Localization of hidden Chua’s attractors,” Phys. Lett. A375, 2230-2233. · Zbl 1242.34102
[20] Leonov, G., Kuznetsov, N. & Vagaitsev, V. [2012] “ Hidden attractor in smooth Chua systems,” Physica D241, 1482-1486. · Zbl 1277.34052
[21] Leonov, G. A. & Kuznetsov, N. V. [2013] “ Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits,” Int. J. Bifurcation and Chaos23, 1330002-1-69. · Zbl 1270.34003
[22] Leonov, G., Kuznetsov, N. & Mokaev, T. [2015a] “ Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Commun. Nonlin. Sci. Numer. Simul.28, 166-174. · Zbl 1510.37063
[23] Leonov, G., Kuznetsov, N. & Mokaev, T. [2015b] “ Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” Eur. Phys. J.224, 1421-1458.
[24] Li, R. & Chen, W. [2013] “ Fractional-order systems without equilibria,” Chin. Phys. B22, 040503.
[25] Molaie, M., Jafari, S., Sprott, J. C. & Golpayegani, S. M. R. H. [2013] “ Simple chaotic flows with one stable equilibrium,” Int. J. Bifurcation and Chaos23, 1350188-1-7. · Zbl 1284.34064
[26] Munoz-Pacheco, J. M., Gómez-Pavón, L. D. C., Félix-Beltrán, O. G. & Luis-Ramos, A. [2013] “ Determining the Lyapunov spectrum of continuous-time 1D and 2D multiscroll chaotic oscillators via the solution of \(<mml:math display=''inline`` overflow=''scroll``>\)-PWL variational equations,” Abstr. Appl. Anal.2013, 851970. · Zbl 1470.34125
[27] Petráš, I. [2006] “ Method for simulation of the fractional-order chaotic systems,” Acta Montanistica Slovaca11, 273-277.
[28] Pezeshki, C., Elgar, S. & Krishna, R. [1990] “ Bispectral analysis of possessing chaotic motion,” J. Sound Vibr.137, 357-368. · Zbl 1235.74189
[29] Pham, V.-T., Jafari, S., Volos, C., Wang, X. & Golpayegani, S. M. R. H. [2014a] “ Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria,” Int. J. Bifurcation and Chaos24, 1450146-1-6. · Zbl 1304.34078
[30] Pham, V.-T., Volos, C., Jafari, S., Wang, X. & Vaidyanathan, S. [2014b] “ Hidden hyperchaotic attractor in a novel simple memristive neural network,” Optoelectron. Adv. Mater.-Rapid Commun.8, 1157-1163.
[31] Pham, V.-T., Volos, C., Jafari, S., Wei, Z. & Wang, X. [2014c] “ Constructing a novel no-equilibrium chaotic system,” Int. J. Bifurcation and Chaos24, 1450073-1-6. · Zbl 1296.34114
[32] Pham, V.-T., Jafari, S., Volos, C., Giakoumis, A., Vaidyanathan, S. & Kapitaniak, T. [2016a] “ A chaotic system with equilibria located on the rounded square loop and its circuit implementation,” IEEE Trans. Circuits Syst.-II63, 878-882.
[33] Pham, V.-T., Jafari, S., Volos, C., Vaidyanathan, S. & Kapitaniak, T. [2016b] “ A chaotic system with infinite equilibria located on a piecewise linear curve,” Optik127, 9111-9117.
[34] Pham, V.-T., Jafari, S., Wang, X. & Ma, J. [2016c] “ A chaotic system with different shapes of equilibria,” Int. J. Bifurcation and Chaos26, 1650069-1-5. · Zbl 1338.34085
[35] Pham, V.-T., Jafari, S. & Volos, C. [2017] “ A novel chaotic system with heart-shaped equilibrium and its circuital implementation,” Optik131, 343-349.
[36] Rajagopal, K., Guessas, L., Karthikeyan, A., Srinivasan, A. & Adam, G. [2017a] “ Fractional-order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional-order PID synchronization,” Complexity2017, 1892618-1-19. · Zbl 1367.93128
[37] Rajagopal, K., Guessas, L., Vaidyanathan, S., Karthikeyan, A. & Srinivasan, A. [2017b] “ Dynamical analysis and FPGA implementation of a novel hyperchaotic system and its synchronization using adaptive sliding mode control and genetically optimized PID control,” Math. Probl. Eng.2017, 7307452-1-14. · Zbl 1426.93143
[38] Rajagopal, K., Karthikeyan, A. & Srinivasan, A. K. [2017c] “ FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization,” Nonlin. Dyn.87, 2281-2304.
[39] Rajagopal, K., Jafari, S., Kacar, S., Karthikeyan, A. & Akgül, A. [2019] “ Fractional-order simple chaotic oscillator with saturable reactors and its engineering applications,” Inf. Tech. Contr.48, 115-128.
[40] Rashtchi, V. & Nourazar, M. [2015] “ FPGA implementation of a real-time weak signal detector using a Duffing oscillator,” Circuits Syst. Sign. Process.34, 3101-3119.
[41] Sharma, P., Shrimali, M., Prasad, A., Kuznetsov, N. & Leonov, G. [2015a] “ Control of multistability in hidden attractors,” Eur. Phys. J.224, 1485-1491.
[42] Sharma, P. R., Shrimali, M. D., Prasad, A., Kuznetsov, N. V. & Leonov, G. A. [2015b] “ Controlling dynamics of hidden attractors,” Int. J. Bifurcation and Chaos25, 1550061-1-7. · Zbl 1314.34134
[43] Sun, H., Abdelwahab, A. & Onaral, B. [1984] “ Linear approximation of transfer function with a pole of fractional power,” IEEE Trans. Autom. Contr.29, 441-444. · Zbl 0532.93025
[44] Tahir, F. R., Jafari, S., Pham, V.-T., Volos, C. & Wang, X. [2015] “ A novel no-equilibrium chaotic system with multiwing butterfly attractors,” Int. J. Bifurcation and Chaos25, 1550056-1-11.
[45] Tavazoei, M. [2007] “ Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems,” IET Sign. Process.1, 171-181.
[46] Tlelo-Cuautle, E., Carbajal-Gomez, V., Obeso-Rodelo, P., Rangel-Magdaleno, J. & Nuñez-Perez, J. C. [2015a] “ FPGA realization of a chaotic communication system applied to image processing,” Nonlin. Dyn.82, 1879-1892. · Zbl 1441.94006
[47] Tlelo-Cuautle, E., Rangel-Magdaleno, J., Pano-Azucena, A., Obeso-Rodelo, P. & Nuñez-Perez, J. C. [2015b] “ FPGA realization of multiscroll chaotic oscillators,” Commun. Nonlin. Sci. Numer. Simul.27, 66-80. · Zbl 1457.37052
[48] Tlelo-Cuautle, E., Pano-Azucena, A., Rangel-Magdaleno, J., Carbajal-Gomez, V. & Rodriguez-Gomez, G. [2016] “ Generating a 50-scroll chaotic attractor at \(66<mml:math display=''inline`` overflow=''scroll``><mml:mspace width=''.17em``>\) Mhz by using FPGAs,” Nonlin. Dyn.85, 2143-2157.
[49] Trzaska, Z. [2011] “ Matlab solutions of chaotic fractional-order circuits,” Engineering Education and Research Using MATLAB (Intech, Rijeka).
[50] Vaidyanathan, S. & Volos, C. [2015] “ Analysis and adaptive control of a novel 3D conservative no-equilibrium chaotic system,” Arch. Contr. Sci.25, 333-353. · Zbl 1446.93044
[51] Wang, Q., Yu, S., Li, C., Lü, J., Fang, X., Guyeux, C. & Bahi, J. M. [2016] “ Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems,” IEEE Trans. Circuits Syst.-I63, 401-412. · Zbl 1469.94118
[52] Wei, Z., Moroz, I. & Liu, A. [2014] “ Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium,” Turkish J. Math.38, 672-687. · Zbl 1401.34054
[53] Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] “ Determining Lyapunov exponents from a time series,” Physica D16, 285-317. · Zbl 0585.58037
[54] Zhang, R. & Gong, J. [2014] “ Synchronization of the fractional-order chaotic system via adaptive observer,” Syst. Sci. Contr. Engin.2, 751-754.
[55] Zhou, Y., Wang, J. & Zhang, L. [2016] Basic Theory of Fractional Differential Equations (World Scientific).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.