time-delayed feedback control of complex nonlinear systems
DESCRIPTION
Net-Works 2008 Pamplona 10.6.2008. TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS. Eckehard Schöll. Institut für Theoretische Physik and Sfb 555 “Complex Nonlinear Processes” Technische Universität Berlin Germany. - PowerPoint PPT PresentationTRANSCRIPT
TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS
Eckehard Schöll
Institut für Theoretische Physikand
Sfb 555 “Complex Nonlinear Processes”Technische Universität Berlin
Germany
http://www.itp.tu-berlin.de/schoell
Net-Works 2008 Pamplona 10.6.2008
OutlineOutline
Introduction: Time-delayed feedback controlTime-delayed feedback control of nonlinear systems
control of deterministic statescontrol of noise-induced oscillations application: lasers, semiconductor nanostructures
Neural systems:Neural systems: control of coherence of neurons and control of coherence of neurons and synchronization of coupled neuronssynchronization of coupled neurons
delay-coupled neuronsdelay-coupled neurons delayed self-feedbackdelayed self-feedback
Control of excitation pulses in Control of excitation pulses in spatio-temporal systemsspatio-temporal systems:: migraine, stroke migraine, stroke non-local instantaneous feedbacknon-local instantaneous feedback time-delayed feedback time-delayed feedback
Why is delay interesting in dynamics?Why is delay interesting in dynamics?
Delay increases the dimension of a differential equation to infinity:
delay generates infinitely many eigenmodes
Delay has been studied in Delay has been studied in classical control theoryclassical control theory and and mechanical engineeringmechanical engineering for a long time for a long time
Simple equation produces very Simple equation produces very complexcomplex behavior behavior
Delay is ubiquitousDelay is ubiquitous
mechanical systems: inertia
electronic systems: electronic systems: capacitive effects capacitive effects ((=RC)=RC) latency time latency time due to processingdue to processing
biological systems: biological systems: cell cycle timecell cycle time biological clocksbiological clocks
neural networks: neural networks: delayed coupling, delayed feedbackdelayed coupling, delayed feedback
optical systems: optical systems: signal transmission timessignal transmission times travelling waves + reflectionstravelling waves + reflections
laser coupled to external cavity (Fabry-laser coupled to external cavity (Fabry-Perot)Perot)multisection lasermultisection lasersemiconductor optical amplifier (SOA)semiconductor optical amplifier (SOA)
Time delayed feedback control methodsTime delayed feedback control methods
Originally invented for controlling chaos (Pyragas 1992): stabilize unstable periodic orbits embedded in a chaotic attractor
More general: More general: stabilization of stabilization of unstable periodic or unstable periodic or stationary statesstationary states in nonlinear dynamic systems in nonlinear dynamic systems
Application to Application to spatio-temporal patterns:spatio-temporal patterns: Partial differential equationsPartial differential equations
Delay can Delay can induce or suppressinduce or suppress instabilities instabilities deterministic delay differential equationsdeterministic delay differential equationsstochastic delay differential equationsstochastic delay differential equations
PublishedOctober 2007
Scope has considerably widened
Time-delayed feedback control Time-delayed feedback control of deterministic systemsof deterministic systems
Time-delayed feedback (Pyragas 1992):Time-delayed feedback (Pyragas 1992):
Stabilisation of unstable periodic orbits Stabilisation of unstable periodic orbits or unstable fixed points or space-time patterns or unstable fixed points or space-time patterns
Time-delay autosynchronisation(TDAS)
Extended time-delay autosynchronisation(ETDAS) (Socolar et al 1994)
)()1((0
txtxRK
)}()({ txtxK
deterministic chaosdeterministic chaosdeterministic chaosdeterministic chaos
=T=T=T=T
Many other schemes
Time-delayed feedback control of deterministic systemsTime-delayed feedback control of deterministic systems
stability is measured byFloquet exponent : x ~ exp(t)or Floquet multiplier =exp(T)
b complex
(1 - )
Beyond Odd Number LimitationBeyond Odd Number Limitation
Example of all-optical time-delayed Example of all-optical time-delayed feedback control in semiconductor laserfeedback control in semiconductor laser
Optical feedback:Optical feedback:
latencynlatency
ni
nin
n
nib
ntt
tEetEeRKetE
,,
)()()(
00
10
||
Stabilisationof fixed point:Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)
Laser: excitable unit, may be coupled to others to form network motif
Stabilization of cw emission:Stabilization of cw emission:Domain of control of unstable fixed pointDomain of control of unstable fixed point
above Hopf bifurcation above Hopf bifurcation
||
Schikora, Hövel, Wünsche, Schöll, Henneberger , PRL 97, 213902 (2006)
Generic model:
phase sensitive coupling
Generic model:
phase sensitive coupling
=0.5T0 =0.9T0
Experimental realizationExperimental realization
||
Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)
Control of spatio-temporal patterns:Control of spatio-temporal patterns: semiconductor nanostructuresemiconductor nanostructure
Without control:Without control:
Examples: Chemical reaction-diffusion systemsChemical reaction-diffusion systemsElectrochemical systemsElectrochemical systemsSemiconductor nanostructuresSemiconductor nanostructuresHodgkin-Huxley neural modelsHodgkin-Huxley neural models
rJuUdt
tdu
x
aaD
xuaf
t
txa
0
1)(
)(),(),(
||
a(x,t): activator variableu(t): inhibitor variable f(a,u): bistable kinetic function D(a): transverse diffusion coefficient
Global coupling:Ratio of timescales:
L
dxuajL
J0
),(1
R
DBRTI
I totU 0
C
U
● Global coupling due to Kirchhoff equation:
jdxUURdt
dUC 0
1 I
Control parameters: = RC, U0
Chaotic breathing pattern
j
u
9.1
u min , u m
ax
= 9.1: above period doubling cascade
Spatially inhomogeneous chaotic oscillations
J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)
Stabilisation of unstable period-1 orbit
u min , u m
ax
●Period doubling bifurcations generate a family of unstable periodic orbits (UPOs)
● Period-1 orbit:
Breathing oscillationsBreathing oscillations
Resonant tunneling diodeResonant tunneling diodea(x,t): electron concentrationa(x,t): electron concentration in quantum well in quantum well u(t): voltage across diodeu(t): voltage across diode
tracking
Time-delayed feedback control Time-delayed feedback control of noise-induced oscillations of noise-induced oscillations
Stabilisation of UPOStabilisation of UPO
noise-inducednoise-inducedoscillationsoscillations
noise-inducednoise-inducedoscillationsoscillations
??
no deterministic orbits!no deterministic orbits!
)}()({ txtxK )}()({ txtxK
deterministic chaosdeterministic chaosdeterministic chaosdeterministic chaos
=T=T=T=T
K. Pyragas, Phys. Lett. A 170, 421 (1992)K. Pyragas, Phys. Lett. A 170, 421 (1992) N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)
Time-delayed feedback control of injection laser with Fabry-Perot resonator
Suppression of noise-induced relaxations oscillations in semiconductor lasers
||
Lang-Kobayashi model:Power spectral densityof optical intensity
Suppression of noisefor 0.5TRO
Flunkert and Schöll,PRE 76, 066202 (2007)
))()(()(1)(
),()(),(),(
0
tutuKtDrJuUdt
tdu
txDx
aaD
xuaf
t
txa
u
a
Feedback control of noise-induced space-time patterns in the DBRT nanostructure
G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)
=4, K=0.4 Du = 0.1, Da = 10-4
Enhancement of temporal regularity:correlation time vs. noise amplitude
vs. feedback gain
=7: increase=7: increase=5: decrease=5: decrease
G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)
Large effect for small noise intensity
Du = 0.1, Da = 10-4
Coherence resonanceCoherence resonance
0
)( dsstcor – – normalized normalized autocorrelation functionautocorrelation function
Correlation time:Correlation time:
Simplified FitzHugh-Nagumo (FHN) system: excitable neuron Simplified FitzHugh-Nagumo (FHN) system: excitable neuron
Excitable SystemExcitable SystemExcitable SystemExcitable System
a=1.1a=1.1=0.01=0.01
Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993)Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993)Pikovsky, Kurths, PRL 78, 775 (1997)Pikovsky, Kurths, PRL 78, 775 (1997)
Example of coherence resonance: neuronExample of coherence resonance: neuron
Simulation from Simulation from S.-G. LeeS.-G. Lee, A., A. Neiman Neiman, , S. KimS. Kim, ,
PREPRE 57, 3292 57, 3292 ( (19981998).).
Time series of the membrane potentialTime series of the membrane potential for for various noise intensityvarious noise intensity::
FitzHugh-Nagumo model with delay FitzHugh-Nagumo model with delay
)()(3
3
tDyyKaxy
yx
xx
)()(3
3
tDyyKaxy
yx
xx
12.4;1.1;01.0 a 12.4;1.1;01.0 a
Janson, Balanov, Schöll, PRL 93, 010601 (2004)
Excitabilitya=1: excitabilitythreshold
u activator (membrane voltage) v inhibitor (recovery variable) time-scale ratio
Coherence vs. Coherence vs. and K and K
D=0.09D=0.09D=0.09D=0.09
D=0.09; K=0.2D=0.09; K=0.2D=0.09; K=0.2D=0.09; K=0.2
Numerics: Balanov, Janson, Schöll, Physica D 199, 1 (2004)Analytics: Prager, Lerch, Schimansky-Geier, Schöll, J. Phys.A 40, 11045 (2007)
2 coupled FitzHugh-Nagumo systems:coupled neuron model as network motif
● 2 non-identical stochastic oscillators: diffusive coupling
frequencies tuned by D1 , D2
B. Hauschildt, N. Janson, A. Balanov, E. Schöll, PRE 74, 051906 (2006)
a= 1.05, 1=0.005, 2= 0.1, D2=0.09 : coherence resonance as function of D1
Stochastic synchronization
● Frequency synchronization : mean interspike intervals (ISI)
● Phase synchronization: 1:1 synchronization index
(Rosenblum et al 2001)
oX+
+ weakly synchronizedo moderately synchronizedx strongly synchronized
Local delayed feedback control: enhance or suppress synchronization
● Moderately synchronized system (o)
System 1
1:1 synchronization index
Delayed coupling, no self-feedback + noise
Dahlem,Hiller, Panchuk,Schöll, IJBC in print, 2008
inducesantiphaseoscillations
Sustained oscillations induced by delayed coupling
excitability parameter a=1.3
a=1.05
Regime of oscillations
excitability parameter a=1.3
Delayed coupling and delayed self-feedback
excitability parameter a=1.3,oscillatory regime,C=K=0.5
Average phase synchronization time:
Schöll, Hiller,Hövel, Dahlem,2008
Spreading depolarization wave(cortical spreading depression SD)
● migraine aura (visual halluzinations)● stroke
Examples:
Migraine aura: neurological precursor(spatio-temporal pattern on visual cortex)
Migraine aura: visual halluzinations
Migraine aura: visual halluzinations
Migraine aura: visual halluzinations
Migraine aura: visual halluzinations
Migraine aura: visual halluzinations
Migraine aura: visual halluzinations
Measured cortical spreading depression
Visual cortex
3 mm/ min
FitzHugh-Nagumo (FHN) system with FitzHugh-Nagumo (FHN) system with activator diffusionactivator diffusion
u activator (membrane voltage) v inhibitor (recovery variable)Du diffusion coefficient time-scale ratio of inhibitor and activator variables excitability parameter
Dahlem, Schneider, Schöll, Chaos (2008)
_
Transient excitation: tissue at risk (TAR)pulses die out after some distance
Dahlem, Schneider, Schöll, J. Theor. Biol. 251, 202 (2008)
different values of and
Boundary of propagation of traveling excitation pulses (SD)
excitable:traveling pulses
non-excitable: transient
Propagation verlocitypulse
FHN system with feedback
Non-local, time-delayed feedback:
Instantaneous long-range feedback:
Time-delayed local feedback:
(electrophysiological activity)
(neurovascular coupling)
Dahlem et alChaos (2008)
Non-local feedback: suppression of CSD
uu
vvuv
vu
Tissue at risk
Non-local feedback:shift of propagation boundary
K=+/-0.2
pulse width x
Time-delayed feedback: suppression of SD
uu vu
uv vv
Tissue at risk
Time-delayed feedback:shift of propagation boundary
uu vu
vv vu
K=+/-0.2
pulse width t
Conclusions
Delayed feedback control of excitable systemsControl of coherence and spectral properties
Stabilization of chaotic deterministic patterns
2 coupled neurons as network motif FitzHugh-Nagumo system: suppression or enhancement of
stochastic synchronization by local delayed feedbackModulation by varying delay timeDelay-coupled neurons:
delay-induced antiphase oscillations of tunable frequency delayed self-feedback: synchronization of oscillation modes
Failure of feedback as mechanism of spreading depression
non-local or time-delayed feedback suppresses propagation of excitation pulses for suitably chosen spatial connections or
time delays
Students
● Roland Aust● Thomas Dahms● Valentin Flunkert● Birte Hauschildt● Gerald Hiller● Johanne Hizanidis● Philipp Hövel● Niels Majer● Felix Schneider
CollaboratorsAndreas AmannAlexander BalanovBernold FiedlerNatalia JansonWolfram JustSylvia SchikoraHans-Jürgen Wünsche
Markus Dahlem
Postdoc
PublishedOctober 2007