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Plan

Plan.  PMF - Skopje  Primeri nelinearnih oscilatora  Fazni prelaz kod modela Kuramoto  Nestabilne fiksne ta~ke i wihova stabilizacija  Nau~na produkcija na Balkanu. PMF, Skopje. Prose~na golemina na evropski oddel za fizika (2009).

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Plan

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  1. Plan  PMF - Skopje  Primeri nelinearnih oscilatora  Fazni prelaz kod modela Kuramoto  Nestabilne fiksne ta~ke i wihova stabilizacija  Nau~na produkcija na Balkanu

  2. PMF, Skopje

  3. Prose~na golemina na evropski oddel za fizika (2009) Studenti - 467 (univerzitet - 23260) Nastaven personal - 79 (univ - 1990) Doktoranti - 75 Na PMF, soodvetno st. 20-30, n. 23 i d. 7-8 . . .

  4. Current programme – part 1(semesters 1-4) (lectures + tutorials + laboratory = credit points) I II Mechanics 4+2+2=8 Molecular physics 4+2+2=8 Mathematical Analysis 1 4+4+0=8 Mathematical analysis 2 3+3+0=7 Computer usage in physics 2+0+2=4 Chemistry 3+0+3=6 Introduction to metrology 2+0+2=4Elective course 3 3+0+0=3 Elective course 1 3+0+0=3 Elective course 4 3+0+0=3 Elective course 2 3+0+0=3 Elective course 5 3+0+0=3 III IV Electromagnetism 4+2+2=7 Optics 4+2+2=8 Mathematical physics 1 3+3+0=7 Mathematical physics 2 3+3+0=7 Theoretical mechanics 3+2+0=6 Electronics 3+1+3=7 Oscillations and waves 2+2+0=4 Theoretical electrodynamics and Elective course 6 3+0+0=3 special theory of relativity 3+2+0=5 Elective course 7 3+0+0=3 Elective course 8 3+0+0=3

  5. Current programme - part 2(semesters 5-8, physics teachers branch) V VI Atomic physics 4+2+2=8 Nuclear physics 4+2+2=8 Measurements in physics 3+0+3=6 Introduction to quantum theory 3+2+0=6 General astronomy 2+1+0=4Introduction to materials 2+0+2=5 Elective course 9 3+0+0=3Basics of solid state physics 3+1+2=6 Elective course 10 3+0+0=3 Pedagogy 3+2+0=5 Elective course 11 3+0+0=3 Elective course 12 3+0+0=3 VII VIII Use of computers in teaching 2+0+2=5Methodology of physics teaching 2 Methodology of physics teaching 1 2+2+3=8 (school practice)2+2+3=8 School experiments 1 2+0+3=6 School experiments 2 2+0+3=5 Psychology 3+2+0=5 Design of electronic equipment 2+0+3=4 Macedonian language 0+2+0=2 History and philosophy of physics 3+1+0=4 Introduction to biophysics 2+0+2=4 Diploma thesis 0+0+9=9

  6. Nonlinear oscillator

  7. The Lorenz system E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20 (1963) 130. Fixed points: C0 (0,0,0) C± (±8.485, ±8.485,27) Eigenvalues: l(C0) = {-22.83, 11.83, -2.67} l(C±) = {-13.85, 0.09+10.19i, 0.09-10.19i} Chaotic attractor of the unperturbed system (F(t)=0)

  8. van der Pol oscillator

  9. Limit cycle

  10. Rössler oscillator with harmonic forcing

  11. 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)

  12. Further examples • The Moon facing the Earth; Gallilean satelites; Kirkwood gaps • Cyclotron and other accelerators • Stroboscope; Fax-machine • Biological clocks; Jet lag • Pacemakers • Farmacological actions of steroids

  13. Further examples 2 • Cardiorespiratory system • Entrainment of cardial and locomotor rhythms • Cardiovascular coupling during anesthesia • Synchronization between parts of the brain • Magnetoencephalographic fields and muscle activity of Parkinsonian patients

  14. Modelot na Kuramoto

  15. Parametar na poredok i sinhronizacija

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

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

  18. Applications Delays are natural in many systems • Coupled oscillators • Electronic circuits • Lasers, electrochemistry • Networks of oscillators • Brain and cardiac dynamics

  19. VARIABLE DELAY FEEDBACK CONTROL OF USS Pyragas control force: - noninvasive for USS and periodic orbits VDFC force: - piezoelements, noise - saw tooth wave: - random wave: - sine wave: A. Gjurchinovski and V. Urumov – Europhys. Lett. 84, 40013 (2008)

  20. VARIABLE DELAY FEEDBACK CONTROL OF USS

  21. THE MECHANISM OF VDFC

  22. DELAY MODULATIONS

  23. THE MECHANISM OF VDFC

  24. THE MECHANISM OF VDFC 2D UNSTABLE FOCUS WITH A DIAGONAL COUPLING n – sufficiently large original system : comparison system : Characteristic equation of the comparison system (2D focus):

  25. THE MECHANISM OF VDFC TDAS VDFC VDFC VDFC

  26. THE MECHANISM OF VDFC The effect of including variable delay into TDAS for small e • condition for the roots lying on the imaginary axis for e=0 to move to • the left half-plane as e increases from zero CONCLUSION: the stability domain will expand in all directions within the half-space K>K0, as soon as e is increased from zero, independent of the precise way in which the delay is varied

  27. THE MECHANISM OF VDFC 2D unstable focus with l = 0.1 and w = p e = 0 (Pyragas) Increase of the stability domain for small e > 0 e = 0 (brown) e = 0.07 (green) e = 0.1 (yellow)

  28. THE MECHANISM OF VDFC e-Kdiagrams for a saw tooth wave modulation (T0=1)

  29. THE MECHANISM OF VDFC

  30. THE MECHANISM OF VDFC Stability analysis for the Lorenz system (saw tooth wave) C0 (0,0,0) C+ (8.485, 8.485,27) C- (-8.485, -8.485,27) s = 10, r = 28, b = 8/3

  31. THE MECHANISM OF VDFC

  32. THE MECHANISM OF VDFC The Rössler system (sawtooth wave) e = 0 e = 0.5 e = 1 e = 2 O.E. Rössler, Phys. Lett. A 57, 397 (1976). Fixed points: C1 (0.007,-0.035,0.035) C2 (5.693, -28.465,28.465) Eigenvalues: l(C1) = {-5.687,0.097+0.995i,0.097-0.995i} l(C2) = {0.192,-0.00000459+5.428i, -0.00000459-5.428i}

  33. T(t) T(t) 2T0 2T0 T0 T0 4t t t 2t 2t 3t 4t 3t t t K(t) K K/2 4t t 2t 3t t STABILIZATION OF UPO BY VDFC • SQUARE WAVE MODULATION • periodic change of the delay, e. g. between T0 and 2T0, K fixed (VDFC) • periodic change of the delay, K varied (VDFC + SCHUSTER, STEMMLER) • - half-period of the wave (optimal choice: t=T0) +

  34. STABILIZATION OF UPO BY VDFC RösslerT0=5.88 • PYRAGAS F(t)=K [y(t-T0)-y(t)] • VDFC (square wave) F(t)=K [y(t-T(t))-y(t)] • SCHUSTER, STEMMLER F(t)=K(t) [y(t-T0)-y(t)] • VDFC (square wave) + SCH-ST F(t)=K(t) [y(t-T(t))-y(t)]

  35. STABILIZATION OF UPO BY VDFC RösslerT0=17.5 RösslerT0=11.75

  36. STABILIZATION OF UPO BY VDFC K periodically varied between K and K/4 (Rössler, T0=17.5) • VDFC + SCHUSTER • Restricted VDFC + SCHUSTER F(t)=K(t) Sin [y(t-T(t))-y(t)]

  37. STABILIZATION OF UPO BY VDFC Rössler T0=5.88 VDFC (square wave) t = T0 t = 2T0 t = T0/2

  38. STABILITY ANALYSIS - RDDE Retarded delay-differential equations • 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) • MOTIVATION:extension of the delay method to infinite dimensional systems • 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 T. Erneux, Applied Delay Differential Equations (Springer, New York, 2009)

  39. DELAY-DIFFERENTIAL EQUATIONS Retarded delay-differential equations General scalar RDDE system: T1≥ 0 – constant delay time F – arbitrary nonlinear function of the state variable x Linearized system around the fixed point x*: 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)

  40. 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) T2 – nominal delay value f– periodic function with zero mean – amplitude of the modulation – frequency of the modulation

  41. STABILITY ANALYSIS - RDDE Stability of the unperturbed system

  42. STABILITY ANALYSIS - RDDE Stability under variable-delay feedback control Limitation of the VDFC for RDDE systems: • A kind of analogue to the odd-number limitation in the case of delayed feedback control of systems described by ordinary differential equations: • W. Just et al., Phys. Rev. Lett. 78, 203(1997) • H. Nakajima, Phys. Lett. A 232, 207 (1997) • … refuted recently: • B. Fiedler et al., Phys. Rev. Lett. 98, 114101 (2007). • B. Fiedler et al., Phys. Rev. E 77, 066207 (2008).

  43. STABILITY ANALYSIS - RDDE Representation of the control boundaries parametrized by  = Im() (K,T2) plane:

  44. EXAMPLES AND SIMULATIONS Mackey-Glass system • A model for regeneration of blood cells in patients with leukemia • M. C. Mackey and L. Glass, Science 197, 28 (1977). • M-G system under variable-delay feedback control: • For the typical valuesa = 0.2, b = 0.1 and c = 10, the fixed points of the free-running system are: • x1 = 0 – unstable for any T1, cannot be stabilized by VDFC • x2 = +1 – stable for T1 [0,4.7082) • x3 = -1 – stable for T1 [0,4.7082)

  45. EXAMPLES AND SIMULATIONS Mackey-Glass system (without control) • T1 = 4 • T1 = 8 • T1 = 15 • T1 = 23

  46. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) T1 = 23 •  = 0 (TDFC) •  = 0.5 (saw) •  = 1 (saw) •  = 2 (saw)

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