| Title:
|
Dependence of hidden attractors on non-linearity and Hamilton energy in a class of chaotic system (English) |
| Author:
|
Zhang, Ge |
| Author:
|
Wang, Chunni |
| Author:
|
Alsaedi, Ahmed |
| Author:
|
Ma, Jun |
| Author:
|
Ren, Guodong |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
54 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
648-663 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Non-linearity is essential for occurrence of chaos in dynamical system. The size of phase space and formation of attractors are much dependent on the setting of nonlinear function and parameters. In this paper, a three-variable dynamical system is controlled by different nonlinear function thus a class of chaotic system is presented, the Hamilton function is calculated to find the statistical dynamical property of the improved dynamical systems composed of hidden attractors. The standard dynamical analysis is confirmed in numerical studies, and the dependence of attractors and Hamilton energy on non-linearity selection is discussed. It is found that lower average Hamilton energy can be detected when intensity of nonlinear function is enhanced. It indicates that non-linearity can decrease the energy cost triggering for dynamical behaviors. (English) |
| Keyword:
|
Helmholtz theorem |
| Keyword:
|
chaos |
| Keyword:
|
hidden attractor |
| Keyword:
|
bifurcation |
| Keyword:
|
Hamilton energy |
| MSC:
|
37B25 |
| MSC:
|
37L30 |
| idZBL:
|
Zbl 06987027 |
| idMR:
|
MR3863249 |
| DOI:
|
10.14736/kyb-2018-4-0648 |
| . |
| Date available:
|
2018-10-30T14:37:56Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147417 |
| . |
| Reference:
|
[1] Ahmad, W. M., Sprott, J. C.: Chaos in fractional-order autonomous nonlinear systems..Chaos 6 (2003), 339-351. 10.1016/s0960-0779(02)00438-1 |
| Reference:
|
[2] Aihara, K., Takabe, T., Toyoda, M.: Chaotic neural networks..Phys. Lett. A 144 (2001), 333-340. MR 1045128, 10.1016/0375-9601(90)90136-c |
| Reference:
|
[3] Aram, Z., Jafari, S., al., J. Ma et: Using chaotic artificial neural networks to model memory in the brain..Commun. Nonlinear Sci. Numer. Simulat. 44 (2017), 449-459. MR 3554829, 10.1016/j.cnsns.2016.08.025 |
| Reference:
|
[4] Bao, B. C., Xu, J. P., Liu, Z.: Initial state dependent dynamical behaviors in a Memristor based chaotic circuit..Chinese Phys. Lett. 27 (2010), 070504. 10.1088/0256-307x/27/7/070504 |
| Reference:
|
[5] Barati, K., Jafari, S., al., J. C. Sprott et: Simple chaotic flows with a curve of equilibria..Int. J. Bifurcat. Chaos 26 (2016), 1630034. MR 3574802, 10.1142/s0218127416300342 |
| Reference:
|
[6] Barrow-Green, J.: Poincaré and the three body problem..Amer. Math. Soc. 2 (1997). MR 1415387 |
| Reference:
|
[7] Chua, L.: Memristor-the missing circuit element..IEEE Trans. Circ. Theory 18 (1971), 507-519. 10.1109/tct.1971.1083337 |
| Reference:
|
[8] Dantsev, D.: A Novel type of chaotic attractor for quadratic systems without equilibriums..Int. J. Bifurcat. Chaos 12 (2002), 659-661. 10.1142/s0218127402004620 |
| Reference:
|
[9] Ditto, W. L., Rauseo, S. N., Spano, M. L.: Experimental control of chaos..Phys. Rev. Lett. 65 (1991), 3211-3214. 10.1103/physrevlett.65.3211 |
| Reference:
|
[10] Dudkowski, D., Jafari, S., al., T. Kapitaniak et: Hidden attractors in dynamical systems..Phys. Rep. 637 (2016), 1-50. MR 3510463, 10.1016/j.physrep.2016.05.002 |
| Reference:
|
[11] Ermakov, I. V., Kingni, S. T., al., V. Z. Tronciu et: Chaotic semiconductor ring lasers subject to optical feedback: Applications to chaos-based communications..Optics Commun. 286 (2013), 265-272. 10.1016/j.optcom.2012.08.063 |
| Reference:
|
[12] Feigenbaum, M. J.: The onset spectrum of turbulence..Phys. Lett. A 74 (1979), 375-378. MR 0591635, 10.1016/0375-9601(79)90227-5 |
| Reference:
|
[13] Garfinkel, A., Spano, M. L., al., W. L. Ditto et: Controlling cardiac chaos..Science 257 (1992), 1230-1235. 10.1126/science.1519060 |
| Reference:
|
[14] Gotthans, T., Petržela, J.: New class of chaotic systems with circular equilibrium..Nonlinear Dyn. 81 (2015), 1141-1149. MR 3367144, 10.1007/s11071-015-2056-7 |
| Reference:
|
[15] Gotthans, T., Sprott, J. C., Petržela, J.: Simple Chaotic Flow with Circle and Square Equilibrium..Int. J. Bifurcat. Chaos 26 (2016), 1650137. MR 3533673, 10.1142/s0218127416501376 |
| Reference:
|
[16] Guo, Y. L., Qi, G. Y., Hamam, Y.: A multi-wing spherical chaotic system using fractal process..Nonlinear Dyn. 85 (2016), 2765-2775. MR 3367161, 10.1007/s11071-016-2861-7 |
| Reference:
|
[17] Hu, X., Liu, C., al., L. Liu et: Multi-scroll hidden attractors in improved Sprott A system..Nonlinear Dyn. 86 (2016), 1725-1734. 10.1007/s11071-016-2989-5 |
| Reference:
|
[18] Itoh, M., Chua, L. O.: Memristor oscillators..Int. J. Bifurcat. Chaos 8 (2008), 3183-3206. MR 2487909, 10.1142/s0218127408022354 |
| Reference:
|
[19] Jafari, M. A., Mliki, E., al., A. Akgul et: Chameleon: the most hidden chaotic flow..Nonlinear Dyn. 88 (2017), 2303-2317. MR 3650512, 10.1007/s11071-017-3378-4 |
| Reference:
|
[20] Jafari, S., Sprott, J. C.: Simple chaotic flows with a line equilibrium..Chaos Solutons Fractals 57 (2013), 79-84. Zbl 1355.37056, MR 3128600, 10.1016/j.chaos.2013.08.018 |
| Reference:
|
[21] Jafari, S., Sprott, J. C., al., V. T. Pham et: Simple chaotic 3D flows with surfaces of equilibria..Nonlinear Dyn. 86 (2016), 1349-1358. 10.1007/s11071-016-2968-x |
| Reference:
|
[22] Jia, B., Gu, H. G., al., L. Li et: Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns..Cogn. Neurodyn. 6 (2012), 89-106. 10.1007/s11571-011-9184-7 |
| Reference:
|
[23] Kennedy, M. P.: Chaos in the Colpitts oscillator..IEEE Trans. Circ. Syst. I 41 (1994), 711-774. 10.1109/81.331536 |
| Reference:
|
[24] Kobe, D. H.: Helmholtz's theorem revisited..Amer. J. Physics 54 (1986), 552-554. 10.1119/1.14562 |
| Reference:
|
[25] Kwok, H. S., Tang, W. K. S.: A fast image encryption system based on chaotic maps with finite precision representation..Chaos Solitons Fractals 32 (2007), 1518-1529. MR 2286314, 10.1016/j.chaos.2005.11.090 |
| Reference:
|
[26] Leonov, G. A., Kuznetsov, N. V., Vagaitsev, V. I.: Hidden attractor in smooth Chua systems..Physica D 241 (2012), 1482-1486. MR 2957820, 10.1016/j.physd.2012.05.016 |
| Reference:
|
[27] Leonov, G. A., Kuznetsov, N. V., Vagaitsev, V. I.: Localization of hidden Chua's attractors..Phys. Lett. A 375 (2011), 2230-2233. Zbl 1242.34102, MR 2800438, 10.1016/j.physleta.2011.04.037 |
| Reference:
|
[28] Leonov, G. A., Kuznetsov, N. V., Vagaitsev, V. I.: Hidden attractor in smooth Chua systems..Physica D 241 (2012), 1482-1486. MR 2957820, 10.1016/j.physd.2012.05.016 |
| Reference:
|
[29] Leonov, G. A., Kuznetsov, N. V., Mokaev, T. N.: Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity..Commun. Nonlinear Sci. Numer. Simulat. 28 (2015), 166-176. MR 3348101, 10.1016/j.cnsns.2015.04.007 |
| Reference:
|
[30] Li, C., Li, S., al., M. Asim et: On the security defects of an image encryption scheme..Image Vision Computing 27 (2009), 1371-1381. 10.1016/j.imavis.2008.12.008 |
| Reference:
|
[31] Li, Y. Y., G., H., Gu: The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns..Nonlinear Dyn. 87 (2017), 2541-2562. 10.1007/s11071-016-3210-6 |
| Reference:
|
[32] Li, X., Li, C., Lee, I. K.: Chaotic image. 125 (2016), 48-63.. 10.1016/j.sigpro.2015.11.017 |
| Reference:
|
[33] Li, F., Yao, C. G.: The infinite-scroll attractor and energy transition in chaotic circuit..Nonlinear Dyn. 84 (2016), 2305-2315. MR 3504299, 10.1007/s11071-016-2646-z |
| Reference:
|
[34] Li, T. Y., Yorke, J. Y.: Period three implies Chaos..Amer. Math. Monthly 82 (1975), 985-992. MR 0385028, 10.2307/2318254 |
| Reference:
|
[35] Lorenz, E. N.: Deterministic nonperiodic flow..J. Atmospher. Sci. 20 (1963), 130-141. 10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2 |
| Reference:
|
[36] Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation..Neurocomputing 205 (2016), 375-381. 10.1016/j.neucom.2016.05.004 |
| Reference:
|
[37] Lv, M., Wang, C., al., G. Ren et: Model of electrical activity in a neuron under magnetic flow effect..Nonlinear Dyn. 85 (2016), 1479-1490. 10.1007/s11071-016-2773-6 |
| Reference:
|
[38] Ma, J., Wu, X. Y., al., R. T. Chu et: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice..Nonlinear Dyn. 76 (2014), 1951-1962. 10.1007/s11071-014-1260-1 |
| Reference:
|
[39] Ma, J., Li, A. B., al., Z. S. Pu et: A time-varying hyperchaotic system and its realization in circuit..Nonlinear Dyn. 62 (2010), 535-541. 10.1007/s11071-010-9739-x |
| Reference:
|
[40] Ma, J., Mi, L., al., P. Zhou et: Phase synchronization between two neurons induced by coupling of electromagnetic field..Appl. Math. Comput. 307 (2017), 321-328. MR 3632742, 10.1016/j.amc.2017.03.002 |
| Reference:
|
[41] Ma, J., Song, X. L., al., J. Tang et: Wave emitting and propagation induced by autapse in a forward feedback neuronal network..Neurocomputing 167 (2015), 378-389. 10.1016/j.neucom.2015.04.056 |
| Reference:
|
[42] Ma, J., Wu, F., al., W. Jin et: Calculation of Hamilton energy and control of dynamical systems with different types of attractors..Chaos 27 (2017), 481-495. MR 3650956, 10.1063/1.4983469 |
| Reference:
|
[43] Ma, J., Wu, F. Q., al., G. D. Ren et: A class of initials-dependent dynamical systems..Appl. Math. Comput. 298 (2017), 65-76. MR 3582328, 10.1016/j.amc.2016.11.004 |
| Reference:
|
[44] Ma, J., Wu, F., Wang, C.: Synchronization behaviors of coupled neurons under electromagnetic radiation..Int. J. Mod Phys. B 31 (2017), 1650251. MR 3599028, 10.1142/s0217979216502519 |
| Reference:
|
[45] Ma, J., Zhang, A. H., al., Y. F. Xia et: Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems..Appl. Math. Comput. 215 (2010), 3318-3326. MR 2576820, 10.1016/j.amc.2009.10.020 |
| Reference:
|
[46] May, R. M.: Simple mathematical models with very complicated dynamics..Nature 261 (1976), 459-467. Zbl 0527.58025, 10.1038/261459a0 |
| Reference:
|
[47] Molaie, M., Jafari, S., al., J. C. Sprott et: Simple chaotic flows with one stable equilibrium..Int. J. Bifurcat. Chaos (2013), 1350188. MR 3150373, 10.1142/s0218127413501885 |
| Reference:
|
[48] Muthuswamy, B.: Implementing memristor based chaotic circuits..Int. J. Bifurcat. Chaos 20 (2010), 1335-1350. 10.1142/s0218127410026514 |
| Reference:
|
[49] Pham, V- T., Jafari, S., al., X. Wang X et: A chaotic system with different shapes of equilibria..Int. J. Bifurcat. Chaos 26 (2016), 1650069. MR 3494063, 10.1142/s0218127416500693 |
| Reference:
|
[50] Pham, V. T., Volos, C., Jafari, S.: A Chaotic system with different families of hidden attractors..Int. J. Bifurcat. Chaos 26 (2016), 1650139. MR 3533675, 10.1142/s021812741650139x |
| Reference:
|
[51] Piper, J. R., Sprott, J. C.: Simple autonomous chaotic circuit..IEEE Trans. Circ. Syst. II 57 (2010), 730-734. 10.1109/tcsii.2010.2058493 |
| Reference:
|
[52] Qi, G. Y., Chen, G. R.: A spherical chaotic system..Nonlinear Dyn. 81 (2015), 1381-1392. MR 3367161, 10.1007/s11071-015-2075-4 |
| Reference:
|
[53] Ren, G. D., Xu, Y., Wang, C. N.: Synchronization behavior of coupled neuron circuits composed of memristors..Nonlinear Dyn. 88 (2017), 893-901. 10.1007/s11071-015-2075-4 |
| Reference:
|
[54] Ryeu, J. K., Aihara, K., Tsuda, I.: Fractal encoding in a chaotic neural network..Phys. Rev. E 64 (2001), 046202. 10.1103/physreve.64.046202 |
| Reference:
|
[55] Shaw, R.: The dripping faucet as a model chaotic system..Aerial Press, Santa Cruz 1984. MR 1101814 |
| Reference:
|
[56] Song, X. L., Jin, W. Y., Ma, J.: Energy dependence on the electric activities of a neuron..Chinese Phys. B 24 (2015), 604-609. 10.1088/1674-1056/24/12/128710 |
| Reference:
|
[57] Strukov, D. B., Snider, G. S., al., D. R. Stewart et: The missing memristor found..Nature 453( 2008), 80-83. 10.1038/nature06932 |
| Reference:
|
[58] Wang, Z. H., Cang, S. J., al., E. O. Ochola et: A hyperchaotic system without equilibrium..Nonlinear Dyn. 69 (2012), 531-537. MR 2929891, 10.1007/s11071-011-0284-z |
| Reference:
|
[59] Wang, X., Chen, G. R.: Constructing a chaotic system with any number of equilibria..Nonlinear Dyn. 71 (2013), 429-436. MR 3015249, 10.1007/s11071-012-0669-7 |
| Reference:
|
[60] Wang, C., Chu, R., Ma, J.: Controlling a chaotic resonator by means of dynamic track control..Complexity 21 (2015), 370-378. MR 3407876, 10.1002/cplx.21572 |
| Reference:
|
[61] Wang, S., Kuang, J., al., J. Li et: Chaos-based secure communications in a large community..Phys. Rev. E 66 (2002), 065202. 10.1103/physreve.66.065202 |
| Reference:
|
[62] Wang, C. N., Ma, J., al., Y. Liu et: Chaos control, spiral wave formation, and the emergence of spatiotemporal chaos in networked Chua circuits..Nonlinear Dyn. 67 (2012), 139-146. 10.1007/s11071-011-9965-x |
| Reference:
|
[63] Wang, C. N., Wang, Y., Ma, J.: Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem..Acta Physica Sinica 65 (2016), 240501. |
| Reference:
|
[64] Wolf, A., Swift, J. B., al., H. L. Swinney et: Determining Lyapunov exponents from a time series..Physica D 16 (1985), 285-317. MR 0805706, 10.1016/0167-2789(85)90011-9 |
| Reference:
|
[65] Wu, C. W., Chua, L. O.: A simple way to synchronize chaotic systems with applications to secure communication systems..Int. J. Bifurcat. Chaos 3 (1993), 1619-1627. 10.1142/s0218127493001288 |
| Reference:
|
[66] Wu, X. Y., Ma, J., al., L. H. Yuan et: Simulating electric activities of neurons by using PSPICE..Nonlinear Dyn. 75 (2014), 113-126. MR 3144840, 10.1007/s11071-013-1053-y |
| Reference:
|
[67] Yalcin, M. E: Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions..Chaos Solutons Fractals 34 (2007), 1659-1666. 10.1016/j.chaos.2006.04.058 |
| Reference:
|
[68] Yang, T.: A survey of chaotic secure communication systems..Int. J. Comput. Cogn. 2 (2004), 81-130. |
| Reference:
|
[69] Zarei, A.: Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors..Nonlinear Dyn. 81 (2015), 585-605. MR 3355053, 10.1007/s11071-015-2013-5 |
| Reference:
|
[70] Zarei, A., Tavakoli, S.: Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system..Appl. Math. Comput. 291 (2016), 323-339. MR 3534407, 10.1016/j.amc.2016.07.023 |
| . |
