Controlling hidden dynamics and multistability of a class of two-dimensional maps via linear augmentation. (English) Zbl 1461.37052
Summary: This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.
MSC:
| 37G35 | Dynamical aspects of attractors and their bifurcations |
| 37M20 | Computational methods for bifurcation problems in dynamical systems |
| 65P30 | Numerical bifurcation problems |
Keywords:
chaotic map; linear augmentation; bifurcation analysis; hidden attractor; control of multistabilityReferences:
| [1] | Bao, B. C., Jiang, T. & Wang, G. Y. [2017] “ Two memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability,” Nonlin. Dyn.89, 1157-1171. |
| [2] | Bao, B. C., Li, H. Z., Zhu, L., Zhang, X. & Chen, M. [2020] “ Initial-switched boosting bifurcations in 2D hyperchaotic map,” Chaos30, 033107-1-14. · Zbl 1445.37028 |
| [3] | Barati, K., Jafari, S., Sprott, J. C. & Pham, V. [2016] “ Simple chaotic flows with a curve of equilibria,” Int. J. Bifurcation and Chaos26, 1630034-1-8. · Zbl 1352.34015 |
| [4] | Brzeski, P. & Perlikowski, P. [2019] “ Sample-based methods of analysis for multistable dynamical systems,” Arch. Comput. Methods Eng.26, 1515-1545. |
| [5] | Chen, Y. & Yang, Q. [2015] “ A new Lorenz-type hyperchaotic system with a curve of equilibria,” Math. Comput. Simulat.112, 40-55. · Zbl 1540.37054 |
| [6] | Chen, E., Min, L. & Chen, G. [2017] “ Discrete chaotic systems with one-line equilibria and their application to image encryption,” Int. J. Bifurcation and Chaos27, 1750046-1-17. · Zbl 1360.37056 |
| [7] | Dantsev, D. [2018] “ A novel type of chaotic attractor for quadratic systems without equilibriums,” Int. J. Bifurcation and Chaos28, 1850001-1-7. · Zbl 1388.34037 |
| [8] | Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N. V., Leonov, G. A. & Prasad, A. [2016a] “ Hidden attractors in dynamical systems,” Phys. Rep.637, 1-50. · Zbl 1359.34054 |
| [9] | Dudkowski, D., Prasad, A. & Kapitaniak, T. [2016b] “ Perpetual points and periodic perpetual loci in maps,” Chaos26, 103103-1-9. · Zbl 1378.37089 |
| [10] | Faghani, Z., Nazarimehr, F., Jafari, S. & Sprott, J. C. [2020] “ A new category of three-dimensional chaotic flows with identical eigenvalues,” Int. J. Bifurcation and Chaos30, 2050026-1-9. · Zbl 1445.34038 |
| [11] | Feng, Y. & Wei, Z. [2015] “ Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors,” Eur. Phys. J. Special Topics224, 1619-1636. |
| [12] | Fonzin Fozin, T., Kengne, R., Kengne, J., Srinivasan, K., Souffo Tagueu, M. & Pelap, F. B. [2019a] “ Control of multistability in a self-excited memristive hyperchaotic oscillator,” Int. J. Bifurcation and Chaos29, 1950119-1-14. · Zbl 1430.94108 |
| [13] | Fonzin Fozin, T., Leutcho, G. D., Kouanou, A. T., Tanekou, G. B., Kengne, R., Kengne, J. & Pelap, F. B. [2019b] “ Multistability control of hysteresis and parallel bifurcation branches through a linear augmentation scheme,” Z. Naturforsch. A75, 11-21. |
| [14] | Fonzin Fozin, T., Megavarna Ezhilarasu, P., Njitacke Tabekoueng, Z., Leutcho, G. D., Kengne, J., Thamilmaran, K., Mezatio, A. B. & Pelap, F. B. [2019c] “ On the dynamics of a simplified canonical Chua’s oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control,” Chaos29, 113105-1-19. · Zbl 1427.34046 |
| [15] | Gotthans, T., Sprott, J. C. & Petržela, J. [2016] “ Simple chaotic flow with circle and square equilibrium,” Int. J. Bifurcation and Chaos26, 1650137-1-8. · Zbl 1345.34016 |
| [16] | Jafari, S. & Sprott, J. C. [2013] “ Simple chaotic flows with a line equilibrium,” Chaos Solit. Fract.57, 79-84. · Zbl 1355.37056 |
| [17] | Jafari, S., Sprott, J. C. & Golpayegani, S. [2013] “ Elementary chaotic flows with no equilibria,” Phys. Lett. A377, 699-702. · Zbl 1428.34059 |
| [18] | Jafari, S., Sprott, J. C. & Molaie, M. [2016a] “ A simple chaotic flow with a plane of equilibria,” Int. J. Bifurcation and Chaos26, 1650098-1-6. · Zbl 1343.34037 |
| [19] | Jafari, S., Sprott, J. C., Pham, V. T., Volos, C. & Li, C. B. [2016b] “ Simple chaotic 3D flows with surfaces of equilibria,” Nonlin. Dyn.86, 1349-1358. |
| [20] | Jafari, S., Pham, V. T., Moghtadaei, M. & Kingni, S. T. [2016c] “ The relationship between chaotic maps and some chaotic systems with hidden attractors,” Int. J. Bifurcation and Chaos26, 1650211-1-8. · Zbl 1354.37044 |
| [21] | Jahanshahi, H., Shahriari-Kahkeshi, M., Alcaraz, R., Wang, X., Singh, V. P. & Pham, V. T. [2019] “ Entropy analysis and neural network-based adaptive control of a non-equilibrium four-dimensional chaotic system with hidden attractors,” Entropy21, 156-1-15. |
| [22] | Jiang, H. B., Liu, Y., Wei, Z. & Zhang, L. P. [2016a] “ Hidden chaotic attractors in a class of two-dimensional maps,” Nonlin. Dyn.85, 2719-2727. |
| [23] | Jiang, H. B., Liu, Y., Wei, Z. & Zhang, L. P. [2016b] “ A new class of three-dimensional maps with hidden chaotic dynamics,” Int. J. Bifurcation and Chaos26, 1650206-1-13. · Zbl 1352.37054 |
| [24] | Jiang, H. B., Liu, Y., Wei, Z. & Zhang, L. P. [2019] “ A new class of two-dimensional chaotic maps with closed curve fixed points,” Int. J. Bifurcation and Chaos29, 1950094-1-10. · Zbl 1419.39006 |
| [25] | Karnatak, R. [2015] “ Linear augmentation for stabilizing stationary solutions: Potential pitfalls and their application,” PLoS One10, e0142238-1-22. |
| [26] | Kuznetsov, Y. A. [1998] Elements of Applied Bifurcation Theory, 2nd edition (Springer-Verlag, NY). · Zbl 0914.58025 |
| [27] | Lai, Q., Nestor, T., Kengne, J. & Zhao, X. W. [2018] “ Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria,” Chaos Solit. Fract.107, 92-102. · Zbl 1380.34070 |
| [28] | Leonov, G. A. & Kuznetsov, N. V. [2013] “ Hidden attractors in dynamical systems: From hidden oscillation 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 |
| [29] | Leonov, G. A., Kuznetsov, N. V. & Mokaev, T. N. [2015] “ 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 |
| [30] | Li, C. B., Sprott, J. C. & Thio, W. [2014a] “ Bistability in a hyperchaotic system with a line equilibrium,” J. Exp. Theoret. Phys.118, 494-500. |
| [31] | Li, Q., Hu, S., Tang, S. & Zeng, G. [2014b] “ Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation,” Int. J. Circuit Th. Appl.42, 1172-1188. |
| [32] | Liu, W. H., Sun, K. H. & He, S. B. [2017] “ SF-SIMM high-dimensional hyperchaotic map and its performance analysis,” Nonlin. Dyn.89, 2521-2532. |
| [33] | Liu, Y. & Páez Chávez, J. [2017a] “ Controlling coexisting attractors of an impacting system via linear augmentation,” Physica D348, 1-11. · Zbl 1376.93050 |
| [34] | Liu, Y. & Páez Chávez, J. [2017b] “ Controlling multistability in a vibro-impact capsule system,” Nonlin. Dyn.88, 1289-1304. |
| [35] | Liu, Y., Lin, W., Páez Chávez, J. & De Sa, R. [2019] “ Torsional stick-slip vibrations and multistability in drill-strings,” Appl. Math. Model.76, 545-557. · Zbl 1481.74366 |
| [36] | Lin, W., Páez Chávez, J., Liu, Y., Yang, Y. & Kuang, Y. [2020] “ Stick-slip suppression and speed tuning for a drill-string system via proportional-derivative control,” Appl. Math. Model.82, 487-502. |
| [37] | Lorenz, H. W. [1993] Nonlinear Dynamical Economics and Chaotic Motion, 2nd edition (Springer-Verlag, Berlin). · Zbl 0841.90036 |
| [38] | Mobayen, S. [2018] “ Design of novel adaptive sliding mode controller for perturbed Chameleon hidden chaotic flow,” Nonlin. Dyn.92, 1539-1553. · Zbl 1398.93277 |
| [39] | 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, 1350188-1-11. · Zbl 1284.34064 |
| [40] | Panahi, S., Sprott, J. C. & Jafari, S. [2018] “ Two simplest quadratic chaotic maps without equilibrium,” Int. J. Bifurcation and Chaos28, 1850144-1-6. · Zbl 1404.34020 |
| [41] | Pham, V. T., Jafari, S. & Wang, X. [2016] “ A chaotic system with different shapes of equilibria,” Int. J. Bifurcation and Chaos26, 1650069-1-5. · Zbl 1338.34085 |
| [42] | Pham, V. T., Volos, C. & Kapitaniak, T. [2017] Systems with Hidden Attractors from Theory to Realization in Circuits (Springer-Verlag, Berlin). · Zbl 1387.37003 |
| [43] | Sambas, A., Mamat, M., Arafa, A. A., Mahmoud, G. M., Mohamed, M. A. & Sanjaya, W. S. [2019] “ A new chaotic system with line of equilibria: Dynamics, passive control and circuit design,” Int. J. Electr. Comput. Eng.9, 2365-2376. |
| [44] | Sharma, P. R., Sharma, A., Shrimali, M. D. & Prasad, A. [2011] “ Targeting fixed-point solutions in nonlinear oscillators through linear augmentation,” Phys. Rev. E83, 067201-1-4. |
| [45] | Sharma, P. R., Shrimali, M. D., Prasad, A. & Feudel, U. [2013] “ Controlling bistability by linear augmentation,” Phys. Lett. A377, 2329-2332. |
| [46] | Sharma, P. R., Singh, A., Prasad, A. & Shrimali, M. D. [2014] “ Controlling dynamical behavior of drive-response system through linear augmentation,” Eur. Phys. J. Special Topics223, 1531-1539. |
| [47] | Singh, J. P., Roy, B. K. & Jafari, S. [2018] “ New family of 4D hyperchaotic and chaotic systems with quadric surfaces of equilibria,” Chaos Solit. Fract.106, 243-257. · Zbl 1392.34049 |
| [48] | Tabekoueng Njitacke, Z., Sami Doubla, I., Kengne, J. & Cheukem, A. [2020] “ Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse,” Chaos30, 023101-1-16. · Zbl 1432.92024 |
| [49] | Thounaojam, U. S. & Shrimali, M. D. [2018] “ Phase-flip in relay oscillators via linear augmentation,” Chaos Solit. Fract.107, 5-12. · Zbl 1380.34079 |
| [50] | Vaidyanathan, S., Jafari, S., Pham, V. T., Azar, A. T. & Alsaadi, F. E. [2018] “ A 4D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design,” Arch. Contr. Sci.28, 239-254. · Zbl 1440.93133 |
| [51] | Wang, X. & Chen, G. [2012] “ A chaotic system with only one stable equilibrium,” Commun. Nonlin. Sci. Numer. Simul.17, 1264-1272. |
| [52] | Wang, X. & Chen, G. [2013] “ Constructing a chaotic system with any number of equilibria,” Nonlin. Dyn.71, 429-436. |
| [53] | Wang, Z., Sun, W., Wei, Z. & Zhang, S. [2015] “ Dynamics and delayed feedback control for a 3D jerk system with hidden attractor,” Nonlin. Dyn.82, 577-588. · Zbl 1348.34113 |
| [54] | Wang, Z., Ma, J., Cang, S., Wang, Z. & Chen, Z. [2016] “ Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors,” Optik127, 2424-2431. |
| [55] | Wang, X., Pham, V. T. & Volos, C. [2017] “ Dynamics, circuit design, and synchronization of a new chaotic system with closed curve equilibrium,” Complexity2017, 7138971-1-9. · Zbl 1367.34048 |
| [56] | Wang, C. F. & Ding, Q. [2018] “ A new two-dimensional map with hidden attractors,” Entropy322, 22-1-10. |
| [57] | Wei, Z. [2011] “ Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A376, 102-108. · Zbl 1255.37013 |
| [58] | 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,” Turk. J. Math.38, 672-687. · Zbl 1401.34054 |
| [59] | Wei, Z. & Zhang, W. [2014] “ Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium,” Int. J. Bifurcation and Chaos24, 1450127-1-14. · Zbl 1302.34017 |
| [60] | Wei, Z., Akgul, A., Kocamaz, U. E., Moroz, I. & Zhang, W. [2018] “ Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo,” Chaos Solit. Fract.111, 157-168. · Zbl 1392.93017 |
| [61] | Zhang, S., Zeng, Y. C., Li, Z. J., Wang, M. J. & Xiong, L. [2018] “ Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability,” Chaos28, 013113-1-11. · Zbl 1390.37133 |
| [62] | Zhang, L. P., Liu, Y., Wei, Z., Jiang, H. B. & Bi, Q. S. [2020] “ A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors,” Chin. Phys. B29, 060501-1-6. |
| [63] | Zhou, P. & Yang, F. [2014] “ Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points,” Nonlin. Dyn.76, 473-480. · Zbl 1319.37030 |
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