×

Hidden extreme multistability, antimonotonicity and offset boosting control in a novel fractional-order hyperchaotic system without equilibrium. (English) Zbl 1406.34047


MSC:

34A34 Nonlinear ordinary differential equations and systems
34A08 Fractional ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abdelaty, A. M.; Elwakil, A. S.; Radwan, A. G.; Psychalinos, C.; Maundy, B. J., Approximation of the fractional-order Laplacian \(s^\alpha\) as a weighted sum of first-order high-pass filters, IEEE Trans. Circuits Syst.-II, 65, 1114-1118, (2018)
[2] Ahmad, W. M.; Sprott, J. C., Chaos in fractional-order autonomous nonlinear systems, Chaos Solit. Fract., 16, 339-351, (2003) · Zbl 1033.37019
[3] Bandt, C.; Pompe, B., Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88, 174102, (2002)
[4] Bao, B.; Jiang, T.; Xu, Q.; Chen, M.; Wu, H.; Hu, Y., Coexisting infinitely many attractors in active band-pass filter-based memristive circuit, Nonlin. Dyn., 86, 1711-1723, (2016)
[5] Bao, B. C.; Xu, Q.; Bao, H.; Chen, M., Extreme multistability in a memristive circuit, Electron. Lett., 52, 1008-1010, (2016)
[6] Bao, B. C.; Bao, H.; Wang, N.; Chen, M.; Xu, Q., Hidden extreme multistability in memristive hyperchaotic system, Chaos Solit. Fract., 94, 102-111, (2017) · Zbl 1373.34069
[7] Bertsias, P.; Psychalinos, C.; Radwan, A. G.; Elwakil, A. S., High-frequency capacitorless fractional-order CPE and FI emulator, Circuits Syst. Sign. Process., 37, 1-20, (2017)
[8] Borah, M. & Roy, B. K. [2017a] “Can fractional-order coexisting attractors undergo a rotational phenomenon?” ISA Trans., in press.
[9] Borah, M.; Roy, B. K., An enhanced multi-wing fractional-order chaotic system with coexisting attractors and switching hybrid synchronisation with its nonautonomous counterpart, Chaos Solit. Fract., 102, 372-386, (2017) · Zbl 1374.34297
[10] Brzeski, P.; Pavlovskaia, E.; Kapitaniak, T.; Perlikowski, P., Controlling multistability in coupled systems with soft impacts, Int. J. Mech. Sci., 127, 118-129, (2016)
[11] Cafagna, D.; Grassi, G., On the simplest fractional-order memristor-based chaotic system, Nonlin. Dyn., 70, 1185-1197, (2012)
[12] Cafagna, D.; Grassi, G., Elegant chaos in fractional-order system without equilibria, Math. Probl. Engin., 2013, 1-7, (2013) · Zbl 1296.34110
[13] Cafagna, D.; Grassi, G., Chaos in a new fractional-order system without equilibrium points, Commun. Nonlin. Sci. Numer. Simul., 19, 2919-2927, (2014)
[14] Cafagna, D.; Grassi, G., Fractional-order systems without equilibria: The first example of hyperchaos and its application to synchronization, Chin. Phys. B, 24, 224-232, (2015)
[15] Chaudhuri, U.; Prasad, A., Complicated basins and the phenomenon of amplitude death in coupled hidden attractors, Phys. Lett. A, 378, 713-718, (2014) · Zbl 1331.34121
[16] Chen, L.; Pan, W.; Wang, K.; Wu, R.; Machado, J. A. T.; Lopes, A. M., Generation of a family of fractional order hyper-chaotic multi-scroll attractors, Chaos Solit. Fract., 105, 244-255, (2017) · Zbl 1380.34119
[17] Elwakil, A. S.; Agambayev, A.; Allagui, A.; Salama, K. N., Experimental demonstration of fractional-order oscillators of orders 2.6 and 2.7, Chaos Solit. Fract., 96, 160-164, (2017)
[18] Gambuzza, L. V.; Frasca, M.; Fortuna, L.; Ntinas, V.; Vourkas, I.; Sirakoulis, G. C., Memristor crossbar for adaptive synchronization, IEEE Trans. Circuits Syst.-I, 64, 2124-2133, (2017)
[19] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 034101, (2003)
[20] He, S.; Sun, K.; Banerjee, S., Dynamical properties and complexity in fractional-order diffusionless Lorenz system, Eur. Phys. J. Plus, 131, 1-12, (2016)
[21] He, S.; Sun, K.; Wang, H., Solution and dynamics analysis of a fractional-order hyperchaotic system, Math. Meth. Appl. Sci., 39, 2965-2973, (2016) · Zbl 1347.34011
[22] Hens, C. R.; Banerjee, R.; Feudel, U.; Dana, S. K., How to obtain extreme multistability in coupled dynamical systems, Phys. Rev. E, 85, 035202, (2012)
[23] Hens, C.; Dana, S. K.; Feudel, U., Extreme multistability: Attractor manipulation and robustness, Chaos, 25, 167-218, (2015) · Zbl 1374.34219
[24] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. Electroanal. Chem., 33, 253-265, (1971)
[25] Jafari, S.; Golpayegani, S. M. R. H.; Sprott, J. C., Elementary quadratic chaotic flows with no equilibria, Phys. Lett. A, 377, 699-702, (2013)
[26] Jafari, S.; Sprott, J. C., Simple chaotic flows with a line equilibrium, Chaos Solit. Fract., 57, 79-84, (2013) · Zbl 1355.37056
[27] Jafari, S.; Sprott, J. C.; Nazarimehr, F., Recent new examples of hidden attractors, Eur. Phys. J. Special Topics, 224, 1469-1476, (2015)
[28] Jia, H. Y.; Chen, Z. Q.; Qi, G. Y., Topological horseshoe analysis and circuit realization for a fractional-order Lü system, Nonlin. Dyn., 74, 203-212, (2013) · Zbl 1281.34067
[29] Kai, D., The analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248, (2002) · Zbl 1014.34003
[30] Kengne, J.; Njitacke, T. Z.; Kamdoum, T. V.; Nguomkam, N. A., Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit, Chaos, 25, 103126, (2015) · Zbl 1374.94910
[31] Kengne, J.; Njitacke, Z. T.; Fotsin, H. B., Dynamical analysis of a simple autonomous jerk system with multiple attractors, Nonlin. Dyn., 83, 751-765, (2016)
[32] Kengne, J.; Negou, A. N.; Njitacke, Z. T., Antimonotonicity, chaos and multiple attractors in a novel autonomous jerk circuit, Nonlin. Dyn., 27, 1-20, (2017) · Zbl 1370.34086
[33] Khodabakhshi, N.; Vaezpour, S. M.; Baleanu, D., Numerical solutions of the initial value problem for fractional differential equations by modification of the adomian decomposition method, Fract. Calcul. Appl. Anal., 17, 382-400, (2014) · Zbl 1308.34015
[34] Kingni, S. T.; Jafari, S.; Simo, H.; Woafo, P., 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. Plus, 129, 1-16, (2014)
[35] Lai, Q.; Akgul, A.; Zhao, X. W.; Pei, H., Various types of coexisting attractors in a new 4D autonomous chaotic system, Int. J. Bifurcation and Chaos, 27, 1750142-1-14, (2017) · Zbl 1373.34025
[36] Lai, Q.; Nestor, T.; Kengne, J.; Zhao, X. W., Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria, Chaos Solit. Fract., 107, 92-102, (2018) · Zbl 1380.34070
[37] Larrondo, H. A.; González, C. M.; Martín, M. T.; Plastino, A.; Rosso, O. A., Intensive statistical complexity measure of pseudorandom number generators, Physica A, 356, 133-138, (2005)
[38] Leonov, G. A.; Kuznetsov, N. V.; Vagaitsev, V. I., Localization of hidden Chua’s attractors, Phys. Lett. A, 375, 2230-2233, (2011) · Zbl 1242.34102
[39] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos Solit. Fract., 22, 549-554, (2004) · Zbl 1069.37025
[40] Li, H.; Liao, X.; Luo, M., A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation, Nonlin. Dyn., 68, 137-149, (2012) · Zbl 1243.93033
[41] Li, C.; Sprott, J. C., Variable-boostable chaotic flows, Optik, 127, 10389-10398, (2016)
[42] Long, J. S., A speech encryption using fractional chaotic systems, Nonlin. Dyn., 65, 103-108, (2010) · Zbl 1251.94013
[43] Ma, J.; Tang, J., A review for dynamics in neuron and neuronal network, Nonlin. Dyn., 20, 1-10, (2017)
[44] Molaie, M.; Jafari, S.; Sprott, J. C., Simple chaotic flows with one stable equilibrium, Int. J. Bifurcation and Chaos, 23, 1350188-1-7, (2013) · Zbl 1284.34064
[45] Munmuangsaen, B.; Srisuchinwong, B., A new five-term simple chaotic attractor, Phys. Lett. A, 373, 4038-4043, (2009) · Zbl 1234.37030
[46] Patel, M. S.; Patel, U.; Sen, A.; Sethia, G. C.; Hens, C.; Dana, S. K.; Feudel, U.; Showalter, K.; Ngonghala, C. N.; Amritkar, R. E., Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators, Phys. Rev. E, 89, 022918, (2014)
[47] Petráš, I., Fractional-order nonlinear systems: Modeling, analysis and simulation, Comput. Math. Appl., 61, 341-356, (2011) · Zbl 1211.65096
[48] Pham, V.-T.; Akgul, A.; Volos, C.; Jafari, S.; Kapitaniak, T., Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable, AEU — Int. J. Electron. Commun., 78, 134-140, (2017)
[49] Pham, V.-T.; Volos, C.; Jafari, S.; Kapitaniak, T., Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlin. Dyn., 87, 2001-2010, (2017)
[50] Pham, V.-T.; Volos, C.; Jafari, S.; Kapitaniak, T., A novel cubic-equilibrium chaotic system with coexisting hidden attractors: Analysis, and circuit implementation, J. Circuits Syst. Comp., 27, 1850066-1-18, (2017)
[51] Pham, V.-T.; Wang, X.; Jafari, S.; Volos, C.; Kapitaniak, T., From Wang-Chen system with only one stable equilibrium to a new chaotic system without equilibrium, Int. J. Bifurcation and Chaos, 27, 1750097-1-9, (2017) · Zbl 1370.34070
[52] Pisarchik, A. N.; Feudel, U., Control of multistability, Phys. Rep., 540, 167-218, (2014) · Zbl 1357.34105
[53] Rajagopal, K.; Karthikeyan, A.; Srinivasan, A. K., 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, (2017)
[54] Shahzad, M.; Pham, V.-T.; Ahmad, M. A.; Jafari, S.; Hadaeghi, F., Synchronization and circuit design of a chaotic system with coexisting hidden attractors, Eur. Phys. J.: Spec. Top., 224, 1637-1652, (2015)
[55] Shao, S. Y., Non-inductive modular circuit of dislocated synchronization of fractional-order Chua’s system and its application, Acta Phys. Sin., 62, 1-27, (2013)
[56] Sharma, P. R.; Shrimali, M. D.; Prasad, A.; Kuznetsov, N. V.; Leonov, G. A., Controlling dynamics of hidden attractors, Int. J. Bifurcation and Chaos, 25, 1550061-1-7, (2015) · Zbl 1314.34134
[57] Silva, C. P., Shil’nikov’s theorem — A tutorial, IEEE Trans. Circuits Syst.-I, 40, 675-682, (1993) · Zbl 0850.93352
[58] Singh, J. P.; Roy, B. K., The simplest 4D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour, Nonlin. Dyn., 89, 1845-1862, (2017)
[59] Sprott, J. C., A proposed standard for the publication of new chaotic systems, Int. J. Bifurcation and Chaos, 21, 2391-2394, (2011)
[60] Sun, H.; Abdelwahab, A.; Onaral, B., Linear approximation of transfer function with a pole of fractional power, IEEE Trans. Autom. Contr., 29, 441-444, (1984) · Zbl 0532.93025
[61] Tseng, C. C., Design of FIR and IIR fractional order simpson digital integrators, Sign. Process., 87, 1045-1057, (2007) · Zbl 1186.94343
[62] Tsirimokou, G.; Psychalinos, C.; Elwakil, A. S.; Salama, K. N., Experimental behavior evaluation of series and parallel connected constant phase elements, AEU — Int. J. Electron. Commun., 74, 5-12, (2017)
[63] Vaidyanathan, S., Analysis and adaptive synchronization of eight-term 3D polynomial chaotic systems with three quadratic nonlinearities, Eur. Phys. J.: Spec. Top., 223, 1519-1529, (2014)
[64] Vaidyanathan, S.; Volos, C., Analysis and adaptive control of a novel 3D conservative no-equilibrium chaotic system, Arch. Contr. Sci., 25, 333-353, (2015)
[65] Von Bremen, H. F.; Udwadia, F. E.; Proskurowski, W., An efficient QR based method for the computation of Lyapunov exponents, Physica D, 101, 1-16, (1997) · Zbl 0885.65078
[66] Xu, B.; Wang, G.; Shen, Y., A simple meminductor-based chaotic system with complicated dynamics, Nonlin. Dyn., 88, 1-19, (2017)
[67] Zhang, C.; Yu, S., Generation of multi-wing chaotic attractor in fractional order system, Chaos Solit. Fract., 44, 845-850, (2011)
[68] Zhang, X.; Li, Z.; Chang, D., Dynamics, circuit implementation and synchronization of a new three-dimensional fractional-order chaotic system, AEU — Int. J. Electron. Commun., 82, 435-445, (2017)
[69] Zhang, S.; Zeng, Y.; Li, Z., One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics, Chin. J. Phys., 56, 793-806, (2018)
[70] Zhang, S.; Zeng, Y.; Li, Z.; Wang, M.; Xiong, L., Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability, Chaos, 28, 013113, (2018) · Zbl 1390.37133
[71] Zhang, S.; Zeng, Y.; Li, Z.; Wang, M.; Zhang, X.; Chang, D., A novel simple no-equilibrium chaotic system with complex hidden dynamics, Int. J. Dyn. Contr., 23, 1-12, (2018)
[72] Zhou, L.; Wang, C.; Zhang, X.; Yao, W., Various attractors, coexisting attractors and antimonotonicity in a simple fourth-order memristive twin-T oscillator, Int. J. Bifurcation and Chaos, 28, 1850050-1-18, (2018) · Zbl 1391.34084
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.