Simple environment for developing methods of controlling chaos in spatially distributed systems

Łukasz Korus

International Journal of Applied Mathematics and Computer Science (2011)

  • Volume: 21, Issue: 1, page 149-159
  • ISSN: 1641-876X

Abstract

top
The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics.

How to cite

top

Łukasz Korus. "Simple environment for developing methods of controlling chaos in spatially distributed systems." International Journal of Applied Mathematics and Computer Science 21.1 (2011): 149-159. <http://eudml.org/doc/208030>.

@article{ŁukaszKorus2011,
abstract = {The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics.},
author = {Łukasz Korus},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {coupled map lattice; spatiotemporal chaos; system stability; Lyapunov exponents; net direction phase},
language = {eng},
number = {1},
pages = {149-159},
title = {Simple environment for developing methods of controlling chaos in spatially distributed systems},
url = {http://eudml.org/doc/208030},
volume = {21},
year = {2011},
}

TY - JOUR
AU - Łukasz Korus
TI - Simple environment for developing methods of controlling chaos in spatially distributed systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2011
VL - 21
IS - 1
SP - 149
EP - 159
AB - The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics.
LA - eng
KW - coupled map lattice; spatiotemporal chaos; system stability; Lyapunov exponents; net direction phase
UR - http://eudml.org/doc/208030
ER -

References

top
  1. Alsing, P.M., Gavrielides, A. and Kovanis, V. (1994). Using neural networks for controling chaos, Physical Review E 49(2): 1225-1231. 
  2. Andrievskii, B.R. and Fradkov, A.L. (2003). Control of chaos: Methods and applications, Automation and Remote Control 64(5): 673-713. Zbl1107.37302
  3. 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. 
  4. 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. 
  5. Astakhov, V.V., Anishchenko, V.S. and Shabunin, A.V. (1995). Controlling spatiotemporal chaos in a chain of the coupled logostic maps, IEEE Transactions on Circuits and Systems 42(6): 352-357. 
  6. Auerbach, D. (1994). Controlling extended systems of chaotic elements, Physical Review Letters 72(8): 1184-1187. 
  7. 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. 
  8. Beck, O., Amann, A., Scholl, E., Socolar, J.E.S. and Just, W. (2002). Comparison of time-delay feedback schemes for spatiotemporal control of chaos in a reaction-diffusion system with global coupling, Physical Review E 66(1): 1-6. 
  9. Boukabou, A. and Mansouri, N. (2005). Predictive control of higher dimensional chaos, Nonlinear Phenomena in Complex Systems 8(3): 258-265. Zbl1121.93024
  10. Chen, G. and Dong, X. (1993). On feedback control of chaotic continuous-time systems, IEEE Transactions on Circuits and Systems 40(9): 591-601. Zbl0800.93758
  11. Chopard, B., Dupuis, A., Masselot, A. and Luthi, P. (2002). Cellular automata and lattice Boltzmann techniques: An approach to model and simulate complex systems, Advances in Complex Systems 5(2): 103-246. Zbl1090.82030
  12. Chui, S.T. and Ma, K.B. (1982). Nature of some chaotic states for Duffing's equation, Physical Review A 26(4): 2262-2265. 
  13. Córdoba, A., Lemos, M. C. and Jiménez-Morales, F. (2006). Periodical forcing for the control of chaos in a chemical reaction, Journal of Chemical Physics 124(1): 1-6. 
  14. Crutchfield, J.P. and Kaneko, K. (1987). Directions in Chaos. Phenomenology of Spatio-Temporal Chaos, World Scientific Publishing Co., Singapore. 
  15. Dressler, U. and Nitsche, G. (1992). Controlling chaos using time delay coordinates, Physical Review Letters 68(1): 1-4. Zbl1194.37136
  16. 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. 
  17. 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. Zbl1056.92511
  18. Greilich, A. and Markus, M. (2003). Correlation of entropy with optimal pinning density for the control of spatiotemporal chaos, Nonlinear Phenomena in Complex Systems 6(1): 541-546. 
  19. 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. 
  20. 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. 
  21. Jacewicz, P. (2002). Model Analysis and Synthesis of Complex Physical Systems Using Cellular Automata, Ph.D. thesis, University of Zielona Góra, Zielona Góra. Zbl1140.82318
  22. Kaneko, K. (1990). Simulating Physics with Coupled Map Lattices, World Scientific Publishing Co., Singapore. 
  23. Korus, L. (2007). Alternative methods of wave motion modelling, in B. Apolloni, R.J. Howlett and L. Jain (Eds.) Knowledge-Based Intelligent Information and Engineering Systems: KES2007/WIRN2007, Part 1, Lecture Notes in Artificial Intelligence, Vol. 4692, Springer-Verlag, Berlin/Heidelberg, pp. 335-345. 
  24. Kwon, Y.S., Ham, S.W. and Lee, K.K. (1997). Analysis of minimal pinning density for controlling spatiotemporal chaos of a coupled map lattice, Physical Review E 55(2): 2009-2012. 
  25. Langenberg, J., Pfister, G. and Abshagen, J. (2004). Chaos from Hopf bifurcation in a fluid flow experiment, Physical Review E 70(4): 046209. Zbl1186.76305
  26. Mihaliuk, E., Sakurai, T., Chirila, F. and Showalter, K. (2002). Feedback stabilization of unstable propagating waves, Physical Review E 65(6): 065602. 
  27. Ott, E. (2002). Chaos in Dynamical Systems, Cambridge University Press, Cambridge. Zbl1006.37001
  28. Ott, E., Grebogi, C. and Yorke, J.A. (1990). Controlling chaos, Physical Review Letters 64(11): 1196-1199. Zbl0964.37501
  29. Parekh, N., Parthasarathy, S. and Sinha, S. (1998). Global and local control of spatiotemporal chaos in coupled map lattice, Physical Review Letters 81(7): 1401-1404. 
  30. Parmananda, P. (1997). Controlling turbulence in coupled map lattice systems using feedback techniques, Physical Review E 56(1): 239-244. 
  31. Procaccia, I. and Meron, E. (1986). Low-dimensional chaos in surface waves: Theoretical analysis of an experiment, Physical Review A 34(4): 3221-3237. 
  32. Pyragas, K. (2001). Control of chaos via an unstable delayed feedback controller, Physical Review Letters 86(11): 2265-2268. 
  33. Singer, J., Wang, Y. and Haim, H. B. (1991). Controlling a chaotic system, Physical Review Letters 66(9): 1123-1125. 
  34. 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): 016218. 
  35. 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. 
  36. Weimar, J.R. (1997). Simulation with Cellular Automata, Logos Verlang Berlin, Berlin. 
  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(4): 045201. 
  39. Zhu, K. and Chen, T. (2001). Controlling spatiotemporal chaos in coupled map lattice, Physical Review E 63(3): 067201. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.