DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK
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Transcript of DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS BY VARIABLE TIME-DELAY FEEDBACK
DESYNCHRONIZATION OF SYSTEMS OF HINDMARSH-ROSE OSCILLATORS
BY VARIABLE TIME-DELAY FEEDBACK
A. Gjurchinovski1, V. Urumov1 and Z. Vasilkoski2
Sts Cyril and Methodius UniversityFaculty of Natural Sciences and MathematicsInstitute of PhysicsP. O. Box 162, 1000 Skopje, Macedonia
E-Mail: [email protected]
International Conference in Memory of Academician Matey Mateev – Sofia , 2011
1 Institute of Physics, Sts Cyril and Methodius University, Skopje, Macedonia2 Northeastern University, Boston, USA
CONTENTS
I. 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
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 thePyragas 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)
VARIABLE DELAY FEEDBACK CONTROL OF USS
Chaotic attractor of theunperturbed system (F(t)=0)
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:(C0) = {-22.83, 11.83, -2.67}(C±) = {-13.85, 0.09+10.19i, 0.09-10.19i}
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)
VARIABLE DELAY FEEDBACK CONTROL OF USS
THE MECHANISM OF VDFC
TDAS VDFC VDFC VDFC
STABILITY ANALYSIS - RDDE
Retarded delay-differential equationsControlled 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)
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 values a = 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)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (without control)
(a) T1 = 4
(b) T1 = 8
(c) T1 = 15
(d) T1 = 23
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
(a) = 0 (TDFC)
(b) = 0.5 (saw)
(c) = 1 (saw)
(d) = 2 (saw)
T1 = 23
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
T1 = 23, T2 = 18, K = 2, = 2, = 5
saw
sin
sqr
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
Caputo fractional-order derivative:
A.Gjurchinovski, T. Sandev and V. Urumov – J. Phys. A43, 445102 (2010)
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler 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
Kuramoto model of phase oscillators
Solution for the Kuramoto model (1975)
2/
2/
2 )sin(cos
dKrgKrr
solutions
0r i 0r
)0(/2 gK c
KKrg c /1/
)(22
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
Hindmarsh-Rose oscillator
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)
Desynchronisation in systems of coupled oscillators
N=1000, tcont=5000, Kmf=0.08, K=0.15, =72.5
TDFC
VDFC
No control
( = 40, = 10)
Desynchronisation in systems of coupled oscillators
Feedback switched on at t=5000
System of 1000 H-R oscillators
=const=72.5
K=0.0036
Kmf=0.08
Desynchronisation in systems of coupled oscillators
TDFC
VDFC
Mean field time-series
( = 40, = 10)
=72.5
Desynchronisation in systems of coupled oscillators
N=1000, tcont=5000, Kmf=0.08, K=0.15, =116
TDFC
VDFC
No control
( = 40, = 10)
Desynchronisation in systems of coupled oscillators
TDFC
VDFC
Mean field time-series
( = 40, = 10)
=116
Desynchronisation in systems of coupled oscillators
Time-delayed feedback control
Variable delay feedback control
(sine-wave, =40, =10, N=1000)
Suppression coefficient
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
Desynchronisation in systems of coupled oscillators
Multiple-delay feedback control
MDFC with variable delay(sine-wave, =40, =10)
Multiple-delay feedback control (MDFC) – Ahlborn, Parlitz (2004)
K1 = K2 = 0.062
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