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DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK

Sts Cyril and Methodius University Faculty of Natural Sciences and Mathematics Institute of Physics P. O. Box 162, 1000 Skopje, Macedonia. DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK. A. Gjurchinovski 1 , V. Urumov 1 and Z. Vasilkoski 2.

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DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK

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  1. Sts Cyril and Methodius University Faculty of Natural Sciences and Mathematics Institute of Physics P. O. Box 162, 1000 Skopje, Macedonia DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK A. Gjurchinovski1, V. Urumov1 and Z. Vasilkoski2 1 Institute of Physics, Sts Cyril and Methodius University, Skopje, Macedonia 2 Northeastern University, Boston, USA E-Mail: urumov@pmf.ukim.mk International Conference in Memory of Academician Matey Mateev – Sofia , 2011

  2. CONTENTS • Introduction • - Time-delay feedback control • - Variable-delay feedback control • II. Stability of fixed points, periodic orbits • - Ordinary differential equations • - Delay-differential equations • - Fractional-order differential equations • III. Desynchronisation in systems of coupled oscillators • IV. Conclusions

  3. INTRODUCTION Time-delayed feedback control - generalizations • Pyragas 1992 – Feedback proportional to the distance • between the current state and the state one period in • the past (TDAS) • Socolar, Sukow, Gauthier 1994 – Improvement of the • Pyragas scheme by using information from many • previous states of the system – commensurate delays • (ETDAS) • Schuster, Stemmler 1997 – Variable gain • Ahlborn, Parlitz 2004 – Multiple delay feedback with • incommensurate delays (MDFC) • Distributed delays (electrical engineering) • Variable delays (mechanical engineering) • Rosenblum, Pikovsky 2004 – Desynchronization of systems of oscillators with constant delay feedback E. Schoell and H. G. Schuster, eds., Handbook of chaos control 2 ed. (Wiley-VCH, Weinheim, 2008)

  4. VARIABLE DELAY FEEDBACK CONTROL OF USS 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)

  5. 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)

  6. VARIABLE DELAY FEEDBACK CONTROL OF USS

  7. THE MECHANISM OF VDFC TDAS VDFC VDFC VDFC

  8. 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 A.Gjurchinovski, V. Urumov – Physical Review E 81, 016209 (2010)

  9. 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)

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

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

  12. EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC) saw sin sqr T1 = 23, T2 = 18, K = 2,  = 2,  = 5

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

  14. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional Rössler system Caputo fractional-order derivative: A.Gjurchinovski, T. Sandev and V. Urumov – J. Phys. A43, 445102 (2010)

  15. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional Rössler system

  16. FRACTIONAL DIFFERENTIAL EQUATIONS Fractional Rössler system - stability diagrams Variable delay feedback control (sine-wave, =1, =10) Variable delay feedback control (sine-wave, =1, =10) Variable delay feedback control (sine-wave, =1, =10) Time-delayed feedback control Time-delayed feedback control Time-delayed feedback control Time-delayed feedback control

  17. Kuramoto model of phase oscillators

  18. Solution for the Kuramoto model (1975) solutions i

  19. DEEP BRAIN STIMULATION • Delay - deliberately introduced to control pathological synchrony manifested in some diseases • Delay - due to signal propagation • Delay – due to self-feedback loop of neurovascular coupling in the brain

  20. Hindmarsh-Rose oscillator

  21. Desynchronisation in systems of coupled oscillators Hindmarsh - Rose oscillators M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. 92, 114102; Phys. Rev. E 70, 041904 (2004) Global coupling Mean field Delayed feedback control

  22. Desynchronisation in systems of coupled oscillators N=1000, tcont=5000,Kmf=0.08, K=0.15, =72.5 No control TDFC VDFC ( = 40,  = 10)

  23. Desynchronisation in systems of coupled oscillators System of 1000 H-R oscillators Feedback switched on at t=5000 Kmf=0.08 K=0.0036 =const=72.5

  24. Desynchronisation in systems of coupled oscillators Mean field time-series =72.5 TDFC VDFC ( = 40,  = 10)

  25. Desynchronisation in systems of coupled oscillators N=1000, tcont=5000,Kmf=0.08, K=0.15, =116 No control TDFC VDFC ( = 40,  = 10)

  26. Desynchronisation in systems of coupled oscillators Mean field time-series =116 TDFC VDFC ( = 40,  = 10)

  27. Desynchronisation in systems of coupled oscillators Variable delay feedback control (sine-wave, =40, =10, N=1000) Time-delayed feedback control X – Mean field in the absence of feedback Xf – Mean field in the presence of feedback T=145 – average period of the mean field in the absence of feedback Suppression coefficient

  28. Desynchronisation in systems of coupled oscillators Multiple-delay feedback control (MDFC) – Ahlborn, Parlitz (2004) 2 K1 = K2 = 0.06 MDFC with variable delay (sine-wave, =40, =10) Multiple-delay feedback control

  29. CONCLUSIONS AND FUTURE PROSPECTS • Enlarged domain for stabilization of unstable steady states in systems of ordinary/delay/fractional differential equations in comparison with Pyragas method and its generalizations • Agreement between theory and simulations for large frequencies in the delay modulation • 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 • The influence of variable-delay feedback in other systems (neutral DDE, PDE, networks, different oscillators, …) • Experimental verification

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