Viktor Urumov - Time-delay feedback control of nonlinear oscillators

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

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)
Studenti - 467 (univerzitet - 23260)
Nastaven personal - 79 (univ - 1990)
Doktoranti - 75
Na PMF, soodvetno st. 20-30, n. 23 i d. 7-8
. . .

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=4 Elective course 3 3+0+0=3
Elective course 1 3+0+0=3 Elective course 4 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

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=4 Introduction to materials 2+0+2=5
Elective course 9 3+0+0=3 Basics 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=5 Methodology 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

Nonlinear oscillator

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}

van der Pol oscillator

Limit cycle

Rössler oscillator with harmonic forcing

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)

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

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

Modelot na Kuramoto

Parametar na poredok i sinhronizacija

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

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

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

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

DELAY MODULATIONS

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):

THE MECHANISM OF VDFC TDAS VDFC VDFC VDFC

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 > K 0 , as soon as  is increased from zero, independent of the precise way in which the delay is varied

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

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

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

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

STABILIZATION OF UPO BY VDFC
SQUARE WAVE MODULATION
periodic change of the delay, e. g. between T 0 and 2 T 0 , K fixed (VDFC)
periodic change of the delay, K varied (VDFC + SCHUSTER, STEMMLER)
- half-period of the wave
( optimal choice:  T 0 )
+ T(t) T 0 2T 0 t     T(t) T 0 2T 0 t     K(t) K/2 K t    

PYRAGAS
Rössler T 0 =5.88
VDFC (square wave)
SCHUSTER, STEMMLER
VDFC (square wave) + SCH-ST
F(t)=K [y(t-T 0 )-y(t)] F(t)=K [y(t-T(t))-y(t)] F(t)=K(t) [y(t-T 0 )-y(t)] F(t)=K(t) [y(t-T(t))-y(t)]

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

VDFC + SCHUSTER
K periodically varied between K and K/4 (Rössler, T 0 =17.5 )
Restricted VDFC + SCHUSTER
F(t)=K(t) Sin [y(t-T(t))-y(t)]

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

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)

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)

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

STABILITY ANALYSIS - RDDE Stability of the unperturbed system

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).

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

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:
x 1 = 0 – unstable for any T 1 , cannot be stabilized by VDFC
x 2 = +1 – stable for T 1  [0,4.7082)
x 3 = -1 – stable for T 1  [0,4.7082)

EXAMPLES AND SIMULATIONS Mackey-Glass system (without control)
T 1 = 4
T 1 = 8
T 1 = 15
T 1 = 23

EXAMPLES AND SIMULATIONS Mackey-Glass system (VDFC)
 = 0 (TDFC)
 = 0.5 (saw)
 = 1 (saw)
 = 2 (saw)
T 1 = 23

 = 1 (sin)
 = 2 (sin)
 = 1 (sqr)
 = 2 (sqr)
T 1 = 23

 = 0 (TDFC)
 = 2 (saw)
 = 2 (sin)
 = 2 (sqr)
K = 0.5

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

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 T 1
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 x 0 =  /4 , the fixed points of the free-running system are:
x 1 = 3.05708 – stable for T 1  [0, 0.82801)
x 2 = 1.05136 – unstable for any T 1 , cannot be stabilized by VDFC
x 3 = -1.86979 – stable for T 1  [0, 0.54767)

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:
x 2n = 2n  – unstable for any T 1 , cannot be stabilized by VDFC
x 2n+1 = (2n+1)  – stable for T 1  [0,  /2 )

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

FRACTIONAL DIFFERENTIAL EQUATIONS Fractional R ö ssler system

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

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 Feedback switched on at t=5000 System of 1000 H-R oscillators  = const =72.5 K=0.0036 K mf =0.08

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

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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Viktor Urumov - Time-delay feedback control of nonlinear oscillators

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