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Control of Steady States

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Control of Complex Nonlinear Systems with Delay

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Notes

  1. 1.

    What is chaos? It is the order which was destroyed at the creation of the world.

  2. 2.

    In the mathematical literature equilibria denote fixed points or steady states of a dynamics system, not to be confused with equilibrium in the physical literature. All systems considered here are far from thermodynamic equilibrium.

References

  1. Gills Z, Iwata C, Roy R, Schwartz IB, Triandaf I (1992) Tracking unstable steady states: extending the stability regime of a multimode laser system. Phys Rev Lett 69:3169

    Article  ADS  Google Scholar 

  2. Bielawski S, Bouazaoui M, Derozier D, Glorieux P (1993) Stabilization and characterization of unstable steady states in a laser. Phys Rev A 47:3276

    Article  ADS  Google Scholar 

  3. Parmananda P, Rhode MA, Johnson GA, Rollins RW, Dewald HD, Markworth AJ (1994) Stabilization of unstable steady states in an electrochemical system using derivative control. Phys Rev E 49:5007

    Article  ADS  Google Scholar 

  4. Parmananda P, Madrigal R, Rivera M, Nyikos L, Kiss IZ, Gáspár V (1999) Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control. Phys Rev E 59:5266

    Article  ADS  Google Scholar 

  5. Ahlborn A, Parlitz U (2004) Stabilizing unstable steady states using multiple delay feedback control. Phys Rev Lett 93:264101

    Article  ADS  Google Scholar 

  6. Rosenblum MG, Pikovsky AS (2004) Controlling synchronization in an ensemble of globally coupled oscillators. Phys Rev Lett 92:114102

    Article  ADS  Google Scholar 

  7. Just W, Bernard T, Ostheimer M, Reibold E, Benner H (1997) Mechanism of time-delayed feedback control. Phys Rev Lett 78:203

    Article  ADS  Google Scholar 

  8. Just W, Reibold E, Benner H, Kacperski K, Fronczak P, Holyst J (1999) Limits of time-delayed feedback control. Phys Lett A 254:158

    Article  ADS  Google Scholar 

  9. Pyragas K (2002) Analytical properties and optimization of time-delayed feedback control. Phys Rev E 66:26207

    Article  ADS  Google Scholar 

  10. von Loewenich C, Benner H, Just W (2004) Experimental relevance of global properties of time-delayed feedback control. Phys Rev Lett 93:174101

    Article  ADS  Google Scholar 

  11. Balanov AG, Janson NB, Schöll E (2005) Delayed feedback control of chaos: bifurcation analysis. Phys Rev E 71:016222

    Article  ADS  Google Scholar 

  12. Pyragas K (1995) Control of chaos via extended delay feedback. Phys Lett A 206:323

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Hövel P, Schöll E (2005) Control of unstable steady states by time-delayed feedback methods. Phys Rev E 72:046203

    Article  Google Scholar 

  14. Yanchuk S, Wolfrum M, Hövel P, Schöll E (2006) Control of unstable steady states by long delay feedback. Phys Rev E 74:026201

    Article  ADS  MathSciNet  Google Scholar 

  15. Dahms T, Hövel P, Schöll E (2007) Control of unstable steady states by extended time-delayed feedback. Phys Rev E 76:056201

    Article  ADS  MathSciNet  Google Scholar 

  16. Gjurchinovski A, Urumov V (2008) Stabilization of unstable steady states by variable delay feedback control. Europhys Lett 84:40013

    Article  ADS  Google Scholar 

  17. Strogatz SH (1994) Nonlinear dynamics and chaos. Westview Press, Cambridge, MA

    Google Scholar 

  18. Schöll E (2001) Nonlinear spatio-temporal dynamics and chaos in semiconductors. Nonlinear science series, vol 10. Cambridge University Press, Cambridge

    Google Scholar 

  19. Schuster HG, Stemmler MB (1997) Control of chaos by oscillating feedback. Phys Rev E 56:6410

    Article  ADS  Google Scholar 

  20. Wright EM (1949) The linear difference-differential equation with constant coefficients. Proc R Soc Edinb A Math Phys Sci 62:387

    Article  Google Scholar 

  21. Wright EM (1955) A non-linear difference-differential equation. J Reine Angew Math 194:66

    MATH  MathSciNet  Google Scholar 

  22. Bellmann R, Cooke KL (1963) Differential-difference equations. Academic Press, New York

    Google Scholar 

  23. Hale JK (1971) Functional differential equations. Applied mathematical sciences, vol 3. Springer, New York

    Book  Google Scholar 

  24. Asl FM, Ulsoy AG (2003) Analysis of a system of linear delay differential equations. ASME J Dyn Syst Meas Control 125:215

    Article  Google Scholar 

  25. Amann A, Schöll E, Just W (2007) Some basic remarks on eigenmode expansions of time-delay dynamics. Physica A 373:191

    Article  ADS  Google Scholar 

  26. Hövel P (2004) Effects of chaos control and latency in time-delay feedback methods. Master’s thesis, Technische Universität Berlin

    Google Scholar 

  27. Hale JK (1977) Theory of functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  28. Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  29. Just W, Reibold E, Kacperski K, Fronczak P, Holyst JA, Benner H (2000) Influence of stable Floquet exponents on time-delayed feedback control. Phys Rev E 61:5045

    Article  ADS  Google Scholar 

  30. Just W (2000) On the eigenvalue spectrum for time-delayed Floquet problems. Physica D 142:153

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Beck O, Amann A, Schöll E, Socolar JES, Just W (2002) Comparison of time-delayed feedback schemes for spatio-temporal control of chaos in a reaction-diffusion system with global coupling. Phys Rev E 66:016213

    Article  ADS  Google Scholar 

  32. Just W, Popovich S, Amann A, Baba N, Schöll E (2003) Improvement of time-delayed feedback control by periodic modulation: analytical theory of Floquet mode control scheme. Phys Rev E 67:026222

    Article  ADS  Google Scholar 

  33. Janson NB, Balanov AG, Schöll E (2004) Delayed feedback as a means of control of noise-induced motion. Phys Rev Lett 93:010601

    Article  ADS  Google Scholar 

  34. Hinz R (2009) Transient behaviour in systems with time-delayed feedback. Master’s thesis, Technische Universität Berlin

    Google Scholar 

  35. Socolar JES, Sukow DW, Gauthier DJ (1994) Stabilizing unstable periodic orbits in fast dynamical systems. Phys Rev E 50:3245

    Article  ADS  Google Scholar 

  36. Baba N (2001) Stabilisierung instabiler räumlicher Muster durch zeitverzögerte Rückkopplung mit räumlichen Filtern. Master’s thesis, Technische Universität Berlin

    Google Scholar 

  37. Dahms T (2007) Stabilisierung von Fixpunkten durch zeitverzögerte Rückkopplung in Lasern. Master’s thesis, Technische Universität Berlin

    Google Scholar 

  38. Blakely JN, Illing L, Gauthier DJ (2004) Controling fast chaos in delay dynamical systems. Phys Rev Lett 92:193901

    Article  ADS  Google Scholar 

  39. Schikora S, Hövel P, Wünsche HJ, Schöll E, Henneberger F (2006) All-optical noninvasive control of unstable steady states in a semiconductor laser. Phys Rev Lett 97:213902

    Article  ADS  Google Scholar 

  40. Sukow DW, Bleich ME, Gauthier DJ, Socolar JES (1997) Controlling chaos in a fast diode resonator using time-delay autosynchronisation: experimental observations and theoretical analysis. Chaos 7:560

    Article  ADS  Google Scholar 

  41. Just W, Reckwerth D, Reibold E, Benner H (1999) Influence of control loop latency on time-delayed feedback control. Phys Rev E 59:2826

    Article  ADS  Google Scholar 

  42. Hövel P, Socolar JES (2003) Stability domains for time-delay feedback control with latency. Phys Rev E 68:036206

    Article  ADS  Google Scholar 

  43. Wünsche HJ, Schikora S, Henneberger F (2008) Noninvasive control of semiconductor lasers by delayed optical feedback. In: Schöll E, Schuster HG (eds) Handbook of chaos control. Wiley-VCH, Weinheim (second completely revised and enlarged edition)

    Google Scholar 

  44. Nakajima H (1997) On analytical properties of delayed feedback control of chaos. Phys Lett A 232:207

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Nakajima H, Ueda Y (1998) Limitation of generalized delayed feedback control. Physica D 111:143

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E (2007) Refuting the odd number limitation of time-delayed feedback control. Phys Rev Lett 98:114101

    Article  ADS  Google Scholar 

  47. Just W, Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E (2007) Beyond odd number limitation: a bifurcation analysis of time-delayed feedback control. Phys Rev E 76:026210

    Article  ADS  MathSciNet  Google Scholar 

  48. Fiedler B, Yanchuk S, Flunkert V, Hövel P, Wünsche HJ, Schöll E (2008) Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser. Phys Rev E 77:066207

    Article  ADS  MathSciNet  Google Scholar 

  49. Dahms T, Hövel P, Schöll E (2008) Stabilizing continuous-wave output in semiconductor lasers by time-delayed feedback. Phys Rev E 78:056213

    Article  ADS  Google Scholar 

  50. Fischer A, Andersen O, Yousefi M, Stolte S, Lenstra D (2000) Experimental and theoretical study of filtered optical feedback in a semiconductor laser. IEEE J Quantum Electron 36:375

    Article  ADS  Google Scholar 

  51. Rosenblum MG, Pikovsky AS (2004) Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms. Phys Rev E 70:041904

    Article  ADS  MathSciNet  Google Scholar 

  52. Flunkert V, Schöll E (2007) Suppressing noise-induced intensity pulsations in semiconductor lasers by means of time-delayed feedback. Phys Rev E 76:066202

    Article  ADS  Google Scholar 

  53. Schikora S, Wünsche HJ, Henneberger F (2008) All-optical noninvasive chaos control of a semiconductor laser. Phys Rev E 78:025202

    Article  ADS  Google Scholar 

  54. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421

    Article  ADS  Google Scholar 

  55. Yanchuk S (2005) Discretization of frequencies in delay coupled oscillators. Phys Rev E 72:036205

    Article  ADS  Google Scholar 

  56. Yanchuk S, Wolfrum M (2004) Instabilities of stationary states in lasers with long-delay optical feedback. Rep Weierstraß Inst Appl Anal Stoch 962:1

    Google Scholar 

  57. Yanchuk S (2005) Properties of stationary states of delay equations with large delay and applications to laser dynamics. Math Methods Appl Sci 28:363

    Article  MATH  MathSciNet  Google Scholar 

  58. Yanchuk S, Wolfrum M (2005) Synchronous and asynchronous instabilities of two lasers with a long delayed coupling. In: van Campen DH, Lazurko MD, van den Oever WPJM (eds) Proceedings of 5th EUROMECH nonlinear dynamics conference ENOC-2005:Eindhoven. Eindhoven University of Technology, Eindhoven, Netherlands), pp 2069–2073, eNOC Eindhoven (CD ROM). ISBN 90 386 2667 3

    Google Scholar 

  59. Lang R, Kobayashi K (1980) External optical feedback effects on semiconductor injection laser properties. IEEE J Quantum Electron 16:347

    Article  ADS  Google Scholar 

  60. Yanchuk S, Wolfrum M (2005) Instabilities of equilibria of delay-differential equations with large delay. In: In: van Campen DH, Lazurko MD, van den Oever WPJM (eds) Proceedings of 5th EUROMECH Nonlinear Dynamics Conference ENOC-2005:Eindhoven. Eindhoven University of Technology, Eindhoven, Netherlands, pp 1060–1065, eNOC Eindhoven (CD ROM). ISBN 90 386 2667 3

    Google Scholar 

  61. Yanchuk S, Wolfrum M (2008) Destabilization patterns in chains of coupled oscillators. Phys Rev E 77:26212

    Article  ADS  MathSciNet  Google Scholar 

  62. Wolfrum M, Yanchuk S (2006) Eckhaus instability in systems with large delay. Phys Rev Lett 96:220201

    Article  ADS  Google Scholar 

  63. Lepri S, Giacomelli G, Politi A, Arecchi FT (1994) High-dimensional chaos in delayed dynmical-systems. Physica D 70:235

    Article  ADS  MATH  MathSciNet  Google Scholar 

  64. Fiedler B, Flunkert V, Georgi M, Hövel P, Schöl E (2008) Beyond the odd number limitation of time-delayed feedback control. In: Schöll E, Schuster HG (eds) Handbook of chaos control. Wiley-VCH, Weinheim, pp 73–84 (second completely revised and enlarged edition)

    Google Scholar 

  65. Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E (2008) Delay stabilization of rotating waves without odd number limitation. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity, vol 1. Wiley-VCH, Weinheim, pp 53–68

    Google Scholar 

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  1. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Philipp Hövel

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Hövel, P. (2010). Control of Steady States. In: Control of Complex Nonlinear Systems with Delay. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14110-2_3

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