Viktor Urumov - Time-delay feedback control of nonlinear oscillators

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Lecture by prof. dr Viktor Urumov (Faculty of Science and Mathematics, Saint Cyril and Methodius University, Skopje, Macedonia) on June 30, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.

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Viktor Urumov - Time-delay feedback control of nonlinear oscillators

  1. 1. Time-delay feedback control of nonlinear oscillators Viktor Urumov,  PMF, Skopje 30 juni 2010, Niš
  2. 2. Plan <ul><li> PMF - Skopje </li></ul><ul><li> Primeri nelinearnih oscilatora </li></ul><ul><li> Fazni prelaz kod modela Kuramoto </li></ul><ul><li> Nestabilne fiksne ta~ke i wihova stabilizacija </li></ul><ul><li> Nau~na produkcija na Balkanu </li></ul>
  3. 3. PMF, Skopje
  4. 4. Prose~na golemina na evropski oddel za fizika (2009) <ul><li>Studenti - 467 (univerzitet - 23260) </li></ul><ul><li>Nastaven personal - 79 (univ - 1990) </li></ul><ul><li>Doktoranti - 75 </li></ul><ul><li>Na PMF, soodvetno st. 20-30, n. 23 i d. 7-8 </li></ul><ul><li>. . . </li></ul>
  5. 5. Current programme – part 1 (semesters 1-4) <ul><li>( lectures + tutorials + laboratory = credit points ) </li></ul><ul><li>I II </li></ul><ul><li>Mechanics 4+2+2=8 Molecular physics 4+2+2=8 </li></ul><ul><li>Mathematical Analysis 1 4+4+0=8 Mathematical analysis 2 3+3+0=7 </li></ul><ul><li>Computer usage in physics 2+0+2=4 Chemistry 3+0+3=6 </li></ul><ul><li>Introduction to metrology 2+0+2=4 Elective course 3 3+0+0=3 </li></ul><ul><li>Elective course 1 3+0+0=3 Elective course 4 3+0+0=3 </li></ul><ul><li>Elective course 2 3+0+0=3 Elective course 5 3+0+0=3 </li></ul><ul><li> </li></ul><ul><li>III IV </li></ul><ul><li>Electromagnetism 4+2+2=7 Optics 4+2+2=8 </li></ul><ul><li>Mathematical physics 1 3+3+0=7 Mathematical physics 2 3+3+0=7 </li></ul><ul><li>Theoretical mechanics 3+2+0=6 Electronics 3+1+3=7 </li></ul><ul><li>Oscillations and waves 2+2+0=4 Theoretical electrodynamics and </li></ul><ul><li>Elective course 6 3+0+0=3 special theory of relativity 3+2+0=5 </li></ul><ul><li>Elective course 7 3+0+0=3 Elective course 8 3+0+0=3 </li></ul>
  6. 6. Current programme - part 2 (semesters 5-8, physics teachers branch ) <ul><li>V VI </li></ul><ul><li>Atomic physics 4+2+2=8 Nuclear physics 4+2+2=8 </li></ul><ul><li>Measurements in physics 3+0+3=6 Introduction to quantum theory 3+2+0=6 </li></ul><ul><li>General astronomy 2+1+0=4 Introduction to materials 2+0+2=5 </li></ul><ul><li>Elective course 9 3+0+0=3 Basics of solid state physics 3+1+2=6 </li></ul><ul><li>Elective course 10 3+0+0=3 Pedagogy 3+2+0=5 </li></ul><ul><li>Elective course 11 3+0+0=3 </li></ul><ul><li>Elective course 12 3+0+0=3 </li></ul><ul><li>VII VIII </li></ul><ul><li>Use of computers in teaching 2+0+2=5 Methodology of physics teaching 2 </li></ul><ul><li>Methodology of physics teaching 1 2+2+3=8 (school practice) 2+2+3=8 </li></ul><ul><li>School experiments 1 2+0+3=6 School experiments 2 2+0+3=5 </li></ul><ul><li>Psychology 3+2+0=5 Design of electronic equipment 2+0+3=4 </li></ul><ul><li>Macedonian language 0+2+0=2 History and philosophy of physics 3+1+0=4 </li></ul><ul><li>Introduction to biophysics 2+0+2=4 Diploma thesis 0+0+9=9 </li></ul>
  7. 7. Nonlinear oscillator
  8. 8. The Lorenz system Chaotic attractor of the unperturbed system ( F(t)=0 ) E. N. Lorenz, “ Deterministic nonperiodic flow ,” J. Atmos. Sci. 20 (1963) 130. Fixed points: C 0 (0,0,0) C ± (±8.485, ±8.485,27) Eigenvalues:  (C 0 ) = {-22.83, 11.83, -2.67}  (C±) = {-13.85, 0.09+10.19i, 0.09-10.19i}
  9. 9. van der Pol oscillator
  10. 10. Limit cycle
  11. 11. Rössler oscillator with harmonic forcing
  12. 12. Historical example from Biology The glowworms ... Represent another shew, which settle on some Trees, like a fiery cloud, with this surprising circumstance, that a whole swarm of these insects, having taken possession of one Tree, and spread themselves over its branches, sometimes hide their Light all at once, and a moment after make it appear again with the utmost regularity and exactness … Engelbert Kaempfer description from his trip in Siam (1680)
  13. 15. Further examples <ul><li>The Moon facing the Earth; Gallilean satelites; Kirkwood gaps </li></ul><ul><li>Cyclotron and other accelerators </li></ul><ul><li>Stroboscope; Fax-machine </li></ul><ul><li>Biological clocks; Jet lag </li></ul><ul><li>Pacemakers </li></ul><ul><li>Farmacological actions of steroids </li></ul>
  14. 16. Further examples 2 <ul><li>Cardiorespiratory system </li></ul><ul><li>Entrainment of cardial and locomotor rhythms </li></ul><ul><li>Cardiovascular coupling during anesthesia </li></ul><ul><li>Synchronization between parts of the brain </li></ul><ul><li>Magnetoencephalographic fields and muscle activity of Parkinsonian patients </li></ul>
  15. 17. Modelot na Kuramoto                                                  
  16. 18. Parametar na poredok i sinhronizacija
  17. 19. Re{enie na modelot na Kuramoto (1975) re{enija i
  18. 22. INTRODUCTION - THE PYRAGAS CONTROL METHOD - Time-delayed feedback control (TDFC) - Time-delayed autosynchronization (TDAS) K. Pyragas, Phys. Lett. A 170 (1992) 421
  19. 23. Applications <ul><li>Delays are natural in many systems </li></ul><ul><li>Coupled oscillators </li></ul><ul><li>Electronic circuits </li></ul><ul><li>Lasers, electrochemistry </li></ul><ul><li>Networks of oscillators </li></ul><ul><li>Brain and cardiac dynamics </li></ul>
  20. 24. Pyragas control force: VARIABLE DELAY FEEDBACK CONTROL OF USS VDFC force: - saw tooth wave: - sine wave: - random wave: - noninvasive for USS and periodic orbits - piezoelements, noise A. Gjurchinovski and V. Urumov – Europhys. Lett. 84 , 40013 (2008)
  21. 25. VARIABLE DELAY FEEDBACK CONTROL OF USS
  22. 26. THE MECHANISM OF VDFC
  23. 27. DELAY MODULATIONS
  24. 28. THE MECHANISM OF VDFC
  25. 29. THE MECHANISM OF VDFC 2D UNSTABLE FOCUS WITH A DIAGONAL COUPLING original system : comparison system :  – sufficiently large Characteristic equation of the comparison system (2D focus):
  26. 30. THE MECHANISM OF VDFC TDAS VDFC VDFC VDFC
  27. 31. THE MECHANISM OF VDFC The effect of including variable delay into TDAS for small  <ul><li>condition for the roots lying on the imaginary axis for  =0 to move to </li></ul><ul><li>the left half-plane as  increases from zero </li></ul>CONCLUSION: the stability domain will expand in all directions within the half-space K > K 0 , as soon as  is increased from zero, independent of the precise way in which the delay is varied
  28. 32. THE MECHANISM OF VDFC 2D unstable focus with  and    Pyragas  Increase of the stability domain for small   (brown)  (green)  (yellow)
  29. 33. THE MECHANISM OF VDFC     diagrams for a saw tooth wave modulation ( T 0 =1)
  30. 34. THE MECHANISM OF VDFC
  31. 35. THE MECHANISM OF VDFC Stability analysis for the Lorenz system (saw tooth wave) C + (8.485, 8.485,27) C 0 (0,0,0) C - (-8.485, -8.485,27)  10, r  28, b  8/3
  32. 36. THE MECHANISM OF VDFC
  33. 37. THE MECHANISM OF VDFC The Rössler system (sawtooth wave) O.E. Rössler, Phys. Lett. A 57 , 397 (1976). Fixed points: C 1 (0.007,-0.035,0.035) C 2 (5.693, -28.465,28.465) Eigenvalues:  (C 1 ) = {-5.687,0.097+0.995i,0.097-0.995i}  (C 2 ) = {0.192,-0.00000459+5.428i, -0.00000459-5.428i}  0  0.5  1  2
  34. 38. STABILIZATION OF UPO BY VDFC <ul><li>SQUARE WAVE MODULATION </li></ul><ul><li>periodic change of the delay, e. g. between T 0 and 2 T 0 , K fixed (VDFC) </li></ul><ul><li>periodic change of the delay, K varied (VDFC + SCHUSTER, STEMMLER) </li></ul><ul><li>- half-period of the wave </li></ul><ul><li>( optimal choice:  T 0 ) </li></ul>+ T(t) T 0 2T 0 t     T(t) T 0 2T 0 t     K(t) K/2 K t    
  35. 39. STABILIZATION OF UPO BY VDFC <ul><li>PYRAGAS </li></ul>Rössler T 0 =5.88 <ul><li>VDFC (square wave) </li></ul><ul><li>SCHUSTER, STEMMLER </li></ul><ul><li>VDFC (square wave) + SCH-ST </li></ul>F(t)=K [y(t-T 0 )-y(t)] F(t)=K [y(t-T(t))-y(t)] F(t)=K(t) [y(t-T 0 )-y(t)] F(t)=K(t) [y(t-T(t))-y(t)]
  36. 40. STABILIZATION OF UPO BY VDFC Rössler T 0 =11.75 Rössler T 0 =17.5
  37. 41. STABILIZATION OF UPO BY VDFC <ul><li>VDFC + SCHUSTER </li></ul>K periodically varied between K and K/4 (Rössler, T 0 =17.5 ) <ul><li>Restricted VDFC + SCHUSTER </li></ul>F(t)=K(t) Sin [y(t-T(t))-y(t)]
  38. 42. STABILIZATION OF UPO BY VDFC Rössler T 0 =5.88 VDFC (square wave)  = T 0  = 2T 0  = T 0 /2
  39. 43. STABILITY ANALYSIS - RDDE Retarded delay-differential equations <ul><li>GOAL: stabilization of unstable steady states by a variable-delay feedback control in a nonlinear dynamical systems described by a scalar autonomous retarded delay-differential equation (RDDE) </li></ul><ul><li>MOTIVATION: extension of the delay method to infinite dimensional systems </li></ul><ul><li>INTEREST: frequent occurrence of scalar RDDE in numerous physical, biological and engineering models, where the time-delays are natural manifestation of the system’s dynamics </li></ul>T. Erneux, Applied Delay Differential Equations (Springer, New York, 2009)
  40. 44. Retarded delay-differential equations General scalar RDDE system: T 1 ≥ 0 – constant delay time F – arbitrary nonlinear function of the state variable x Linearized system around the fixed point x * : DELAY-DIFFERENTIAL EQUATIONS Characteristic equation for the stability of steady state x * of the free-running system: A. Gjurchinovski and V. Urumov – Phys. Rev. E 81 , 016209 (2010)
  41. 45. STABILITY ANALYSIS - RDDE Retarded delay-differential equations Controlled RDDE system: u(t) – Pyragas-type feedback force with a variable time delay K – feedback gain (strength of the feedback) T 2 – nominal delay value f – periodic function with zero mean  – amplitude of the modulation  – frequency of the modulation
  42. 46. STABILITY ANALYSIS - RDDE Stability of the unperturbed system
  43. 47. STABILITY ANALYSIS - RDDE Stability under variable-delay feedback control Limitation of the VDFC for RDDE systems: <ul><li>A kind of analogue to the odd-number limitation in the case of delayed feedback control of systems described by ordinary differential equations: </li></ul><ul><ul><ul><li>W. Just et al., Phys. Rev. Lett. 78 , 203(1997) </li></ul></ul></ul><ul><ul><ul><li>H. Nakajima, Phys. Lett. A 232 , 207 (1997) </li></ul></ul></ul><ul><li>… refuted recently: </li></ul><ul><ul><ul><li>B. Fiedler et al., Phys. Rev. Lett. 98 , 114101 (2007). </li></ul></ul></ul><ul><ul><ul><li>B. Fiedler et al., Phys. Rev. E 77 , 066207 (2008). </li></ul></ul></ul>
  44. 48. STABILITY ANALYSIS - RDDE Representation of the control boundaries parametrized by  = Im(  ) ( K , T 2 ) plane:
  45. 49. EXAMPLES AND SIMULATIONS Mackey-Glass system <ul><li>A model for regeneration of blood cells in patients with leukemia </li></ul><ul><ul><ul><ul><li>M. C. Mackey and L. Glass, Science 197 , 28 (1977). </li></ul></ul></ul></ul><ul><li>M-G system under variable-delay feedback control: </li></ul><ul><li>For the typical values a = 0.2 , b = 0.1 and c = 10 , the fixed points of the free-running system are: </li></ul><ul><ul><ul><li>x 1 = 0 – unstable for any T 1 , cannot be stabilized by VDFC </li></ul></ul></ul><ul><ul><ul><li>x 2 = +1 – stable for T 1  [0,4.7082) </li></ul></ul></ul><ul><ul><ul><li>x 3 = -1 – stable for T 1  [0,4.7082) </li></ul></ul></ul>
  46. 50. EXAMPLES AND SIMULATIONS Mackey-Glass system (without control) <ul><li>T 1 = 4 </li></ul><ul><li>T 1 = 8 </li></ul><ul><li>T 1 = 15 </li></ul><ul><li>T 1 = 23 </li></ul>
  47. 51. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) <ul><li> = 0 (TDFC) </li></ul><ul><li> = 0.5 (saw) </li></ul><ul><li> = 1 (saw) </li></ul><ul><li> = 2 (saw) </li></ul>T 1 = 23
  48. 52. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) <ul><li> = 1 (sin) </li></ul><ul><li> = 2 (sin) </li></ul><ul><li> = 1 (sqr) </li></ul><ul><li> = 2 (sqr) </li></ul>T 1 = 23
  49. 53. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) <ul><li> = 0 (TDFC) </li></ul><ul><li> = 2 (saw) </li></ul><ul><li> = 2 (sin) </li></ul><ul><li> = 2 (sqr) </li></ul>K = 0.5
  50. 54. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) T 1 = 23, T 2 = 18, K = 2,  = 2,  = 5 saw sin sqr
  51. 55. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC)
  52. 56. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC)
  53. 57. EXAMPLES AND SIMULATIONS Ikeda system <ul><li>Introduced to describe the dynamics of an optical bistable resonator, incorporating the round-trip time of light in an optical cavity via the time delay T 1 </li></ul><ul><ul><ul><ul><li>K. Ikeda, Opt. Commun. 39 , 257 (1979) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>K. Ikeda and K. Matsumoto, Physica D 29 , 223 (1987). </li></ul></ul></ul></ul><ul><li>Ikeda system under variable-delay feedback control: </li></ul><ul><li>For  = 4 and x 0 =  /4 , the fixed points of the free-running system are: </li></ul><ul><ul><ul><li>x 1 = 3.05708 – stable for T 1  [0, 0.82801) </li></ul></ul></ul><ul><ul><ul><li>x 2 = 1.05136 – unstable for any T 1 , cannot be stabilized by VDFC </li></ul></ul></ul><ul><ul><ul><li>x 3 = -1.86979 – stable for T 1  [0, 0.54767) </li></ul></ul></ul>
  54. 58. EXAMPLES AND SIMULATIONS Sprott system <ul><li>The simplest one-parameter RDDE system with a sinusoidal nonlinearity </li></ul><ul><ul><ul><ul><li>J. C. Sprott, Phys. Lett. A 366 , 397 (2007) </li></ul></ul></ul></ul><ul><li>Sprott system under variable-delay feedback control: </li></ul><ul><li>The fixed points of the free-running system are: </li></ul><ul><ul><ul><li>x 2n = 2n  – unstable for any T 1 , cannot be stabilized by VDFC </li></ul></ul></ul><ul><ul><ul><li>x 2n+1 = (2n+1)  – stable for T 1  [0,  /2 ) </li></ul></ul></ul>
  55. 59. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system Caputo fractional-order derivative:
  56. 60. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system
  57. 61. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system - stability diagrams Time-delayed feedback control Variable delay feedback control (sine-wave,  =1,  =10) Time-delayed feedback control Variable delay feedback control (sine-wave,  =1,  =10) Time-delayed feedback control Variable delay feedback control (sine-wave,  =1,  =10) Time-delayed feedback control
  58. 62. Desynchronisation in systems of coupled oscillators Hindmarsh - Rose oscillators Mean field Global coupling Delayed feedback control M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. 92 , 114102; Phys. Rev. E 70 , 041904 (2004)
  59. 63. Desynchronisation in systems of coupled oscillators Feedback switched on at t=5000 System of 1000 H-R oscillators  = const =72.5 K=0.0036 K mf =0.08
  60. 64. Desynchronisation in systems of coupled oscillators Time-delayed feedback control Variable delay feedback control (sine-wave,  =40,  =10) Suppression coefficient X – Mean field in the absence of feedback X f – Mean field in the presence of feedback T=145 – average period of the mean field in the absence of feedback
  61. 65. CONCLUSIONS AND FUTURE PROSPECTS <ul><li>Enlarged domain for stabilization of unstable steady states in systems of ordinary/delay/fractional differential equations in comparison with Pyragas method and its generalizations </li></ul><ul><li>Agreement between theory and simulations for large frequencies in the delay variability </li></ul><ul><li>The enlargement of the control domain may undergo a complex rearrangement depending on the type of the delay modulation </li></ul><ul><li>Extended area of stabilization of periodic orbits by noninvasive variable-delay feedback control </li></ul><ul><li>Variable delay feedback control provides increased robustness in achieving desynchronization in wider domain of parameter space in system of coupled Hindmarsh-Rose oscillators interacting through their mean field </li></ul><ul><li>The influence of variable-delay feedback in other systems (neutral DDE, PDE, networks, …) </li></ul><ul><li>Experimental verification </li></ul>
  62. 66. SCI publikacii od balkanski gradovi 2006-2010 IT, GR, DE, FR, US 7 7 22 147 162 348 Tirana DE, US, FR, IT 72 241 953 760 6826 8964 Sofija DE, BG, US, SRB, IT 15 22 58 520 628 1257 Skopje DE, CRO, US, SRB, SLO 3 9 48 192 565 824 Saraevo SRB, DE, IT, FR, RU 5 13 54 287 363 Podgorica GR, US, UK, DE 25 71 175 162 1354 1858 Nikozija US, DE, IT, UK, FR 87 358 1129 733 7957 10482 Qubqana US, DE, RU, PL 6 23 120 123 768 1044 Ki{inev US, DE, UK, IT, FR 703 443 1031 2772 15135 20627 Istanbul US, DE, IT, FR, SLO 194 373 936 1252 6590 9576 Zagreb FR, DE, US, IT, UK 32 205 1523 1312 8184 11413 Bukure{t DE, US, IT, UK, FR 112 242 860 1669 7287 10348 Belgrad US, UK, DE, FR, IT 1032 1592 1751 4996 16700 26880 Atina glavna sorabotka pisma revijalni zbornici apstrakti statii vkupno
  63. 67. SCI publikacii od Skopje 1993-2009 (Sv. Kiril i Metodij)

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