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: urumov@pmf.ukim.mk

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