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.

1. Time-delay feedback control of nonlinear oscillators Viktor Urumov, PMF, Skopje 30 juni 2010, Niš

3. PMF, Skopje

7. Nonlinear oscillator

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. van der Pol oscillator

10. Limit cycle

11. Rössler oscillator with harmonic forcing

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)

17. Modelot na Kuramoto

18. Parametar na poredok i sinhronizacija

19. Re{enie na modelot na Kuramoto (1975) re{enija i

22. INTRODUCTION - THE PYRAGAS CONTROL METHOD - Time-delayed feedback control (TDFC) - Time-delayed autosynchronization (TDAS) K. Pyragas, Phys. Lett. A 170 (1992) 421

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)

25. VARIABLE DELAY FEEDBACK CONTROL OF USS

26. THE MECHANISM OF VDFC

27. DELAY MODULATIONS

28. THE MECHANISM OF VDFC

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):

30. THE MECHANISM OF VDFC TDAS VDFC VDFC VDFC

32. THE MECHANISM OF VDFC 2D unstable focus with  and    Pyragas  Increase of the stability domain for small   (brown)  (green)  (yellow)

33. THE MECHANISM OF VDFC     diagrams for a saw tooth wave modulation ( T 0 =1)

34. THE MECHANISM OF VDFC

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

36. THE MECHANISM OF VDFC

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

40. STABILIZATION OF UPO BY VDFC Rössler T 0 =11.75 Rössler T 0 =17.5

42. STABILIZATION OF UPO BY VDFC Rössler T 0 =5.88 VDFC (square wave)  = T 0  = 2T 0  = T 0 /2

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)

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

46. STABILITY ANALYSIS - RDDE Stability of the unperturbed system

48. STABILITY ANALYSIS - RDDE Representation of the control boundaries parametrized by  = Im(  ) ( K , T 2 ) plane:

54. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) T 1 = 23, T 2 = 18, K = 2,  = 2,  = 5 saw sin sqr

55. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC)

56. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC)

59. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system Caputo fractional-order derivative:

60. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system

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

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)

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

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

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

67. SCI publikacii od Skopje 1993-2009 (Sv. Kiril i Metodij)