Efficiency analysis of control algorithms in spatially distributed systems with chaotic behavior

Korus, Ł.

International Journal of Applied Mathematics and Computer Science, Tome 24 (2014) no. 4, pp. 759-770.

Résumé

The paper presents results of examination of control algorithms for the purpose of controlling chaos in spatially distributed systems like the coupled map lattice (CML). The mathematical definition of the CML, stability analysis as well as some basic results of numerical simulation exposing complex, spatiotemporal and chaotic behavior of the CML were already presented in another paper. The main purpose of this article is to compare the efficiency of controlling chaos by simple classical algorithms in spatially distributed systems like CMLs. This comparison is made based on qualitative and quantitative evaluation methods proposed in the previous paper such as the indirect Lyapunov method, Lyapunov exponents and the net direction phase indicator. As a summary of this paper, some conclusions which can be useful for creating a more efficient algorithm of controlling chaos in spatially distributed systems are made.

Détail

Keywords: controlling spatiotemporal chaos, coupled map lattice, system stability, Lyapunov exponent, net direction phase
Mots-clés : stabilność układu, wykładnik Lapunowa, kierunek fazy

Bibliographie

[1] Abarbanel, H. (1996). Analysis of Observed Chaotic Data, Springer-Verlag, New York, NY.

[2] Alsing, P.M., Gavrielides, A. and Kovanis, V. (1994). Using neural networks for controlling chaos, Physical Review E 49(2): 1225–1231.

[3] Andrievskii, B.R. and Fradkov, A.L. (2003). Control of chaos: Methods and applications, Automation and Remote Control 64(5): 673–713.

[4] Andrzejak, R.G., Lehnertz, K., Mormann, F., Rieke, C., David, P. and Elger, C.E. (2001). Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state, Physical Review E 64(1): 1–8.

[5] Argoul, F., Arneodo, A., Richetti, P. and Roux, J.C. (1987). From quasiperiodicity to chaos in the Belousov–Zhabotinskii reaction, I: Experiment, Journal of Chemical Physics 86(6): 3325–3339.

[6] Banerjee, S., Misra, A.P., Shukla, P.K. and Rondoni, L. (2010). Spatiotemporal chaos and the dynamics of coupled Langmuir and ion-acoustic waves in plasmas, Physical Review E 81(1): 1–10.

[7] Bashkirtseva, I. and Ryashko, L. (2013). Attainability analysis in the problem of stochastic equilibria synthesis for nonlinear discrete systems, International Journal of Applied Mathematics and Computer Science 23(1): 5–16, DOI: 10.2478/amcs-2013-0001.

[8] Boukabou, A. and Mansouri, N. (2005). Predictive control of higher dimensional chaos, Nonlinear Phenomena in Complex Systems 8(3): 258–265.

[9] Chen, G. and Dong, X. (1993). On feedback control of chaotic continuous-time systems, IEEE Transactions on Circuits and Systems 40(9): 591–601.

[10] Chen, Q., Teng, Z. and Hu, Z. (2013). Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response, International Journal of Applied Mathematics and Computer Science 23(2): 247–261, DOI: 10.2478/amcs-2013-0019.

[11] Chui, S.T. and Ma, K.B. (1982). Nature of some chaotic states for Duffing’s equation, Physical Review A 26(4): 2262–2265.

[12] Cordoba, A., Lemos, M.C. and Jimenez-Morales, F. (2006). Periodical forcing for the control of chaos in a chemical reaction, Journal of Chemical Physics 124(1): 1–6.

[13] Crutchfield, J.P. and Kaneko, K. (1987). Directions in Chaos: Phenomenology of Spatio-Temporal Chaos, World Scientific Publishing Co., Singapore.

[14] Dressler, U. and Nitsche, G. (1992). Controlling chaos using time delay coordinates, Physical Review Letters 68(1): 1–4.

[15] Gautama, T., Mandic, D.P. and Hulle, M.M.V. (2003). Indications of nonlinear structures in brain electrical activity, Physical Review E 67(1): 1–5.

[16] Govindan, R.B., Narayanan, K. and Gopinathan, M.S. (1998). On the evidence of deterministic chaos in ECG: Surrogate and predictability analysis, Chaos 8(2): 495–502.

[17] Gunaratne, G.H., Lisnay, P.S. and Vinson, M.J. (1989). Chaos beyond onset: A comparison of theory and experiment, Physical Review Letters 63(1): 1–4.

[18] Held, G.A., Jeffries, C. and Haller, E.E. (1984). Observation of chaotic behavior in an electron-hole plasma in GE, Physical Review Letters 52(12): 1037–1040.

[19] Kaneko, K. (1990). Simulating Physics with Coupled Map Lattices, World Scientific Publishing Co., Singapore.

[20] Korus, Ł. (2011). Simple environment for developing methods of controlling chaos in spatially distributed systems, International Journal of Applied Mathematics and Computer Science 21(1): 149–159, DOI: 10.2478/v10006-011-0011-4.

[21] Langenberg, J., Pfister, G. and Abshagen, J. (2004). Chaos from Hopf bifurcation in a fluid flow experiment, Physical Review E 70(1): 1–5.

[22] Lasota, A. and Mackey, M. (1997). Chaos, Fractals, and Noise, Springer, New York, NY.

[23] Mitkowski, P.J. and Mitkowski, W. (2012). Ergodic theory approach to chaos: Remarks and computational aspects, International Journal of Applied Mathematics and Computer Science 22(2): 259–267, DOI: 10.2478/v10006-012-0019-4.

[24] Ogorzałek, M.J., Galias, Z., Dabrowski, W. and Dabrowski, A. (1996a). Spatio-temporal co-operative phenomena in CNN arrays composed of chaotic circuits simulation experiments, International Journal of Circuit Theory and Applications 24(3): 261–268.

[25] Ogorzałek, M.J., Galias, Z., Dabrowski, W. and Dabrowski, A. (1996b). Wave propagation, pattern formation and memory effects in large arrays of interconnected chaotic circuits, International Journal of Bifurcation and Chaos 6(10): 1859–1871.

[26] Ott, E. (2002). Chaos in Dynamical Systems, Cambridge University Press, Cambridge.

[27] Ott, E., Grebogi, C. and Yorke, J.A. (1990). Controlling chaos, Physical Review Letters 64(11): 1196–1199.

[28] Parmananda, P. (1997). Controlling turbulence in coupled map lattice systems using feedback techniques, Physical Review E 56(1): 239–244.

[29] Procaccia, I. and Meron, E. (1986). Low-dimensional chaos in surface waves: Theoretical analysis of an experiment, Physical Review A 34(4): 3221–3237.

[30] Pyragas, K. (2001). Control of chaos via an unstable delayed feedback controller, Physical Review Letters 86(11): 2265–2268.

[31] Sauer, T., Yorke, J., and Casdagli, M. (1991). Embedology, Journal of Statistical Physics 65(3–4): 579–616.

[32] Singer, J., Wang, Y. and Haim, H.B. (1991). Controlling a chaotic system, Physical Review Letters 66(9): 1123–1125.

[33] Stark, J. (1999). Delay embeddings for forced systems, I: Deterministic forcing, Jouranl of Nonlinear Science 9(3): 255–332.

[34] Takens, F. (1981). Detecting Strange Attractors in Turbulence, Springer-Verlag, Berlin.

[35] Used, J. and Martin, J.C. (2010). Multiple topological structures of chaotic attractors ruling the emission of a driven laser, Physical Review E 82(1): 1–7.

[36] Wei, W., Zonghua, L. and Bambi, H. (2000). Phase order in chaotic maps and in coupled map lattices, Physical Review Letters 84(12): 2610–2613.

[37] Yamada, T. and Graham, R. (1980). Chaos in a laser system under a modulated external field, Physical Review Letters 45(16): 1322–1324.

[38] Yim, G., Ryu, J., Park, Y., Rim, S., Lee, S., Kye, W. and Kim, C. (2004). Chaotic behaviors of operational amplifiers, Physical Review E 69(1): 1–4.