Generating chaos via decentralized linear state feedback and a class of nonlinear functions. (English) Zbl 1092.37020
Summary: This paper proposes a general chaotification algorithm for a class of nonlinear discrete-time dynamical systems. Based on a given nonlinear discrete-time system, the new chaotification algorithm uses the decentralized linear state feedback control and a class of nonlinear functions that are only required to satisfy some mild assumptions to construct a chaotic nonlinear system. The sine and sawtooth function used in the existing literature on chaotification only are the special cases of the proposed nonlinear function. Based on the corrected version of the Marotto theorem, we mathematically prove that the constructed nonlinear system is indeed chaotic in the sense of Li and Yorke. In particular, an explicit formula for the computation of chaotification parameters is also obtained. The theoretical results in this paper can be used to supervise one to construct different chaotification algorithms.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 93C10 | Nonlinear systems in control theory |
| 93D15 | Stabilization of systems by feedback |
References:
| [1] | Lakshmanan, M.; Murali, K., Chaos in nonlinear oscillators: Controlling and synchronization (1996), World Scientific: World Scientific Singapore · Zbl 0868.58058 |
| [2] | (Judd, K.; Mees, A.; Teo, K. L.; Vincent, T., Control and chaos: Mathematical modeling (1997), Birkhauser: Birkhauser Boston) |
| [3] | Chen, G.; Dong, X., From chaos to order: Perspectives, methodologies, and applications (1998), World Scientific: World Scientific Singapore · Zbl 0908.93005 |
| [4] | Kapitaniak, T., Chaos for engineers: Theory, applications, and control (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0904.58052 |
| [5] | Fradkov, A. L.; Pogromsky, A. Y., Introduction to control of oscillations and chaos (1999), World Scientific: World Scientific Singapore · Zbl 0945.93003 |
| [6] | Chen G, Fradkov AL. Chaos control and synchronization bibliographies (1987-2001). Available from: <http://www.ee.cityu.edu.hk/ gchen/chaos-papers.html; Chen G, Fradkov AL. Chaos control and synchronization bibliographies (1987-2001). Available from: <http://www.ee.cityu.edu.hk/ gchen/chaos-papers.html |
| [7] | Chen, G., Chaos control and anticontrol, IEEE Circuits Syst. Soc. Newsletter, March, 1-5 (1998) |
| [8] | Wang, X. F.; Chen, G., Chaotification via arbitrarily small feedback controls: Theory, method, and applications, Int. J. Bifurcat. Chaos, 10, 549-570 (2000) · Zbl 1090.37532 |
| [9] | Schiff, S. J.; Jerger, K.; Duong, D. H.; Chang, T.; Spano, M. L.; Ditto, W. L., Controlling chaos in the brain, Nature, 370, 615-620 (1994) |
| [10] | Garfinkel, A.; Spano, M. L.; Ditto, W. L.; Weiss, J. N., Controlling cardiac chaos, Science, 257, 1230-1235 (1992) |
| [11] | Jakimoski, G.; Kocarev, L., Chaos and cryptography Block encryption ciphers based on chaotic maps, IEEE Trans. Circuits Syst.-I, 48, 163-169 (2001) · Zbl 0998.94016 |
| [12] | Chen, G.; Lai, D., Feedback control of Lyapunov exponents for discrete-time dynamical systems, Int. J. Bifurcat. Chaos, 6, 1341-1349 (1996) · Zbl 0875.93157 |
| [13] | Chen, G.; Lai, D., Making a dynamical system chaotic: Feedback control of Lyapunov exponents for discrete-time dynamical systems, IEEE Trans. Circuits Syst. I, 44, 250-253 (1997) |
| [14] | Chen, G.; Lai, D., Feedback anticontrol of chaos, Int. J. Bifurcat. Chaos, 8, 1585-1590 (1998) · Zbl 0941.93522 |
| [15] | Wang, X. F.; Chen, G., On feedback anticontrol of discrete-time chaos, Int. J. Bifurcat. Chaos, 9, 1435-1441 (1999) · Zbl 0964.93039 |
| [16] | Li, Z.; Park, J. B.; Joo, Y. H.; Choi, Y. H.; Chen, G., Anticontrol of chaos for discrete TS fuzzy systems, IEEE Trans Circuit Syst. I, 49, 249-253 (2002) · Zbl 1368.93324 |
| [17] | Lu Jun Guo. Yet another algorithm for anticontrol of Li-Yorke chaos by a sinusoidal nonlinearity. Int J Nonlinear Mech, submitted for publication; Lu Jun Guo. Yet another algorithm for anticontrol of Li-Yorke chaos by a sinusoidal nonlinearity. Int J Nonlinear Mech, submitted for publication |
| [18] | Wang, X. F.; Chen, G., Chaotifying a stable LTI system by tiny feedback control, IEEE Trans. Circuit Syst. I, 47, 410-415 (2000) |
| [19] | Li, Z.; Park, J. B.; Chen, G.; Joo, Y. H.; Choi, Y. H., Generating chaos via feedback control from a stable TS fuzzy system through a sinusoidal nonlinearity, Int. J. Bifurcat. Chaos, 12, 2283-2291 (2002) · Zbl 1052.93509 |
| [20] | Zheng, Y.; Chen, G.; Liu, Z., On chaotification of discrete systems, Int. J. Bifurcat. Chaos, 13, 3443-3447 (2003) · Zbl 1057.37082 |
| [21] | Zheng, Y.; Chen, G., Single state-feedback chaotification of discrete dynamical systems, Int. J. Bifurcat. Chaos, 14, 279-284 (2004) · Zbl 1099.37507 |
| [22] | Lu Jun Guo. Chaotification of discrete systems by a class of decentralized nonlinear state feedback control. IEEE Trans Circuit Syst II, submitted for publication; Lu Jun Guo. Chaotification of discrete systems by a class of decentralized nonlinear state feedback control. IEEE Trans Circuit Syst II, submitted for publication |
| [23] | Wang, X. F.; Chen, G.; Yu, X., Anticontrol of chaos in continuous-time systems via time-delay feedback, Chaos, 4, 771-779 (2000) · Zbl 0967.93045 |
| [24] | Zheng, Z. H.; Lü, J. H.; Chen, G.; Zhou, T. S.; Zhang, S. C., Generating two simultaneously chaotic attractors with a switching piecewise-linear controller, Chaos, Solitons & Fractals, 20, 277-288 (2004) · Zbl 1045.37018 |
| [25] | Zhou, T. S.; Chen, G.; Yang, Q. G., Constructing a new chaotic system based on the Silnikov criterion, Chaos, Solitons & Fractals, 19, 985-993 (2004) · Zbl 1053.37015 |
| [26] | Marotto, F. R., Snap-back repellers imply chaos in \(R^n\), J. Math. Anal. Appl., 63, 199-223 (1978) · Zbl 0381.58004 |
| [27] | Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 481-485 (1975) |
| [28] | Devaney, R. L., An introduction to chaotic dynamical system (1987), Addison-Wesley: Addison-Wesley New York |
| [29] | Chen, G.; Hsu, S. B.; Zhou, J., Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span, J. Math. Phys., 39, 6459-6489 (1998) · Zbl 0959.37027 |
| [30] | Lin, W.; Ruan, J.; Zhao, W. R., On the mathematical clarification of the snap-back-repeller in high-dimensional systems and chaos in a discrete neural network model, Int. J. Bifurcat. Chaos, 12, 1129-1139 (2002) · Zbl 1044.37020 |
| [31] | Li, C. P.; Chen, G., An improved version of the Marotto Theorem, Chaos, Solitons & Fractals, 18, 69-77 (2003) · Zbl 1135.37311 |
| [32] | Chen, M.; Chen, Z.; Chen, G., Approximate solutions of operator equations (1997), World Scientific: World Scientific Singapore · Zbl 0914.65062 |
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.
