viktor urumov - time-delay feedback control of nonlinear oscillators
DESCRIPTION
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.TRANSCRIPT
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Plan
PMF - Skopje Primeri nelinearnih oscilatora Fazni prelaz kod modela Kuramoto Nestabilne fiksne ta~ke i wihova
stabilizacija Nau~na produkcija na Balkanu
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PMF, Skopje
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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 . . .
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Current programme – part 1(semesters 1-4)
(lectures + tutorials + laboratory = credit points)
I IIMechanics 4+2+2=8 Molecular physics
4+2+2=8Mathematical 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=6Introduction to metrology 2+0+2=4 Elective course 3 3+0+0=3Elective course 1 3+0+0=3 Elective course 4
3+0+0=3Elective course 2 3+0+0=3 Elective course 5
3+0+0=3
III IVElectromagnetism 4+2+2=7 Optics 4+2+2=8Mathematical physics 1 3+3+0=7 Mathematical physics 2 3+3+0=7Theoretical mechanics 3+2+0=6 Electronics 3+1+3=7Oscillations and waves 2+2+0=4 Theoretical electrodynamics andElective course 6 3+0+0=3 special theory of relativity 3+2+0=5Elective course 7 3+0+0=3 Elective course 8 3+0+0=3
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Current programme - part 2(semesters 5-8, physics teachers branch)
V VIAtomic physics 4+2+2=8 Nuclear physics
4+2+2=8Measurements in physics 3+0+3=6 Introduction to quantum theory 3+2+0=6General astronomy 2+1+0=4 Introduction to materials 2+0+2=5Elective course 9 3+0+0=3 Basics of solid state physics 3+1+2=6Elective course 10 3+0+0=3 Pedagogy 3+2+0=5Elective course 11 3+0+0=3Elective course 12 3+0+0=3
VII VIIIUse of computers in teaching 2+0+2=5 Methodology of physics teaching 2Methodology of physics teaching 1 2+2+3=8 (school practice) 2+2+3=8School experiments 1 2+0+3=6 School experiments 2 2+0+3=5Psychology 3+2+0=5 Design of electronic equipment 2+0+3=4Macedonian language 0+2+0=2 History and philosophy of physics 3+1+0=4Introduction to biophysics 2+0+2=4 Diploma thesis 0+0+9=9
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Nonlinear oscillator
sin sin
sin sin
x b x x A t
x y
y x by A t
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The Lorenz system
Chaotic attractor of theunperturbed system (F(t)=0)
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}
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van der Pol oscillator
2
2
( 1) 0
(1 )
x x x x
x y
y x y x
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Limit cycle
- 2 - 1 0 1 2displacement xHtL- 2
- 1
0
1
2yticolev
xvHtL
b=0.5
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Rössler oscillator with harmonic forcing
sin( )
( )
extx y z E t
y x ay
z f z x c
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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)
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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
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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
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Modelot na Kuramoto
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Parametar na poredok i sinhronizacija
1r 0r
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Re{enie na modelot na Kuramoto (1975)
2/
2/
2 )sin(cos
dKrgKrr
re{enija
0r i 0r
)0(/2 gK c
KKrg c /1/
)(22
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INTRODUCTION - THE PYRAGAS CONTROL METHOD
- Time-delayed feedback control (TDFC)- Time-delayed autosynchronization (TDAS)
K. Pyragas, Phys. Lett. A 170 (1992) 421
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Applications
Delays are natural in many systems
• Coupled oscillators
• Electronic circuits
• Lasers, electrochemistry
• Networks of oscillators
• Brain and cardiac dynamics
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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)
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VARIABLE DELAY FEEDBACK CONTROL OF USS
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THE MECHANISM OF VDFC
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DELAY MODULATIONS
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THE MECHANISM OF VDFC
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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):
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THE MECHANISM OF VDFC
TDAS VDFC VDFC VDFC
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THE MECHANISM OF VDFC
The effect of including variable delay into TDAS for small
• condition for the roots lying on the imaginary axis for =0 to move to the left half-plane as increases from zero
CONCLUSION: the stability domain will expand in all directions within the half-space K>K0, as soon as is increased from zero, independent of the precise way in which the delay is varied
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THE MECHANISM OF VDFC
2D unstable focus withand
Pyragas
Increase of the stability domain for small
(brown)
(green)
(yellow)
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THE MECHANISM OF VDFC
diagrams for a saw tooth wave modulation (T0=1)
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THE MECHANISM OF VDFC
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THE MECHANISM OF VDFC
Stability analysis for the Lorenz system (saw tooth wave)
C+ (8.485, 8.485,27)
C0 (0,0,0)
C- (-8.485, -8.485,27)
10, r 28, b 8/3
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THE MECHANISM OF VDFC
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THE MECHANISM OF VDFC
The Rössler system (sawtooth wave)
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:(C1) = {-5.687,0.097+0.995i,0.097-0.995i}(C2) = {0.192,-0.00000459+5.428i, -0.00000459-5.428i}
0 0.5
1 2
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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)
T(t)
T0
2T0
t
- half-period of the wave (optimal choice: T0)
T(t)
T0
2T0
t
K(t)
K/2
K
t
+
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STABILIZATION OF UPO BY VDFC
•PYRAGASRössler T0=5.88
•VDFC (square wave)
•SCHUSTER, STEMMLER
•VDFC (square wave) + SCH-ST
F(t)=K [y(t-T0)-y(t)]
F(t)=K [y(t-T(t))-y(t)]
F(t)=K(t) [y(t-T0)-y(t)]
F(t)=K(t) [y(t-T(t))-y(t)]
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STABILIZATION OF UPO BY VDFC
Rössler T0=11.75 Rössler T0=17.5
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STABILIZATION OF UPO BY VDFC
•VDFC + SCHUSTER
K periodically varied between K and K/4 (Rössler, T0=17.5)
•Restricted VDFC + SCHUSTER F(t)=K(t) Sin [y(t-T(t))-y(t)]
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STABILIZATION OF UPO BY VDFC
Rössler T0=5.88VDFC (square wave)
= T0
= 2T0
= T0/2
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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)
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Retarded delay-differential equationsGeneral scalar RDDE
system:
T1 ≥ 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)
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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
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STABILITY ANALYSIS - RDDE
Stability of the unperturbed system
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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).
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STABILITY ANALYSIS - RDDE
Representation of the control boundaries parametrized by = Im()
(K,T2) plane:
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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)
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (without control)
(a) T1 = 4
(b) T1 = 8
(c) T1 = 15
(d) T1 = 23
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
(a) = 0 (TDFC)
(b) = 0.5 (saw)
(c) = 1 (saw)
(d) = 2 (saw)
T1 = 23
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
(a) = 1 (sin)
(b) = 2 (sin)
(c) = 1 (sqr)
(d) = 2 (sqr)
T1 = 23
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
(a) = 0 (TDFC)
(b) = 2 (saw)
(c) = 2 (sin)
(d) = 2 (sqr)
K = 0.5
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
T1 = 23, T2 = 18, K = 2, = 2, = 5
saw
sin
sqr
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
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EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
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EXAMPLES AND SIMULATIONS
Ikeda system
• 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 T1
K. Ikeda, Opt. Commun. 39, 257 (1979)K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).
• Ikeda system under variable-delay feedback control:
• For = 4 and x0 = /4, the fixed points of the free-running system are:
• x1 = 3.05708 – stable for T1 [0, 0.82801)• x2 = 1.05136 – unstable for any T1, cannot be stabilized by
VDFC• x3 = -1.86979 – stable for T1 [0, 0.54767)
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EXAMPLES AND SIMULATIONS
Sprott system
• The simplest one-parameter RDDE system with a sinusoidal nonlinearity
J. C. Sprott, Phys. Lett. A 366, 397 (2007)
• Sprott system under variable-delay feedback control:
• The fixed points of the free-running system are:
• x2n = 2n – unstable for any T1, cannot be stabilized by VDFC
• x2n+1 = (2n+1) – stable for T1 [0, /2)
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FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
Caputo fractional-order derivative:
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FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
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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
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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)
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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
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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
Xf – Mean field in the presence of feedback
T=145 – average period of the mean field in the absence of feedback
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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 variability
• The enlargement of the control domain may undergo a complex rearrangement depending on the type of the delay modulation
• Extended area of stabilization of periodic orbits by noninvasive variable-delay feedback control
• 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, …)
• Experimental verification
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SCI publikacii od balkanski gradovi 2006-2010
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SCI publikacii od Skopje 1993-2009(Sv. Kiril i Metodij)
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