Transcript Folie 1 - Universidad de Navarra

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
http://www.itp.tu-berlin.de/schoell
Outline
iIntroduction: Time-delayed feedback control
of nonlinear systems
8control of deterministic states
8control of noise-induced oscillations
8application: lasers, semiconductor nanostructures
i Neural systems: control of coherence of neurons and
synchronization of coupled neurons
8delay-coupled neurons
8delayed self-feedback
iControl of excitation pulses in spatio-temporal systems:
migraine, stroke
8non-local instantaneous feedback
8time-delayed feedback
Why is delay interesting in dynamics?
iDelay increases the dimension of a differential
equation to infinity:
delay t
generates infinitely many eigenmodes
i Simple equation produces very complex behavior
iDelay has been studied in classical control theory
and mechanical engineering for a long time
Delay is ubiquitous
imechanical systems: inertia
ielectronic systems: capacitive effects (t=RC)
latency time due to processing
ioptical systems: signal transmission times
travelling waves + reflections
8laser coupled to external cavity (Fabry-Perot)
8multisection laser
8semiconductor optical amplifier (SOA)
i biological systems: cell cycle time
biological clocks
8 neural networks: delayed coupling, delayed feedback
Time delayed feedback control methods
iOriginally invented for controlling chaos
(Pyragas 1992):
stabilize unstable periodic orbits embedded in a
chaotic attractor
iMore general: stabilization of unstable periodic or
stationary states in nonlinear dynamic systems
iDelay can induce or suppress instabilities
8deterministic delay differential equations
8stochastic delay differential equations
i Application to spatio-temporal patterns:
8 Partial differential equations
Published
October 2007
Scope has considerably widened
Time-delayed feedback control
of deterministic systems
Stabilisation of unstable periodic orbits
or unstable fixed points or space-time patterns
iTime-delayed feedback (Pyragas 1992):
K{x(t  )  x(t )}
=T
Time-delay autosynchronisation
(TDAS)
Extended time-delay
autosynchronisation
(ETDAS) (Socolar et al 1994)

K  R  x( t  (  1)  x( t    )
 0
deterministic chaos
Many other schemes
Time-delayed feedback control of deterministic systems
stability is measured by
Floquet exponent L: dx ~ exp(Lt)
or Floquet multiplier m=exp(LT)
Beyond Odd Number Limitation
b complex
(1 - gl)
Example of all-optical time-delayed
feedback control in semiconductor laser
|
Optical feedback:


Eb ( t )  Ke i  R n e in E ( t n )  e i E ( t n1 )
n 0
   0 latency ,    0 , t n  t  n   latency

Stabilisation
of 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:
Domain of control of unstable fixed point
above Hopf bifurcation
=0.5T0
=0.9T0
|
Generic model:
phase sensitive coupling
l0.2/T0 0.2 /2p
Schikora, Hövel, Wünsche, Schöll, Henneberger , PRL 97, 213902 (2006)
Experimental realization
|
Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)
Control of spatio-temporal patterns:
semiconductor nanostructure
Without control:
a ( x , t )
 
a 
 f ( a , u )   D( a ) 
t
x 
x 
du( t ) 1
 U 0  u  rJ 
dt

a(x,t): activator variable
u(t): inhibitor variable
f(a,u): bistable kinetic function
|
D(a): transverse diffusion
coefficient
L
Global coupling: J  1  j(a , u)dx
Ratio of timescales:  L 0
Examples: Chemical reaction-diffusion systems
Electrochemical systems
Semiconductor nanostructures
Hodgkin-Huxley neural models
●
U
DBRT
I
Global coupling due to Kirchhoff equation:
C
dU
1
C
 U 0  U    jdx
dt
R
Control parameters:  = RC, U0
I
U0
R
I tot
Chaotic breathing pattern
umin , umax
 = 9.1: above period doubling cascade
Spatially inhomogeneous
chaotic oscillations
9.1
j
u
J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)
Stabilisation of unstable period-1 orbit
●
Period doubling
bifurcations generate a
family of unstable
periodic orbits (UPOs)
Period-1 orbit:
tracking
umin , umax
●
Breathing oscillations
Resonant tunneling diode
a(x,t): electron concentration
in quantum well
u(t): voltage across diode
Time-delayed feedback control
of noise-induced oscillations
Stabilisation of UPO
K{x(t  )  x(t )}
=T
deterministic chaos
K. Pyragas, Phys. Lett. A 170, 421 (1992)
no deterministic orbits!
K{x(t  )  x(t )}
?
noise-induced
oscillations
N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)
Suppression of noise-induced relaxations
oscillations in semiconductor lasers
Time-delayed feedback control of injection laser with Fabry-Perot
resonator
|
Lang-Kobayashi model:
Power spectral density
of optical intensity
Suppression of noise
for   0.5 TRO
Flunkert and Schöll,
PRE 76, 066202 (2007)
Feedback control of noise-induced spacetime patterns in the DBRT nanostructure
a ( x , t )
 
a 
 f (a , u)   D(a )   Da ( x , t )
t
x 
x 
du( t ) 1
 U 0  u  rJ   Du ( t )  K ( u( t )  u( t   ))
dt

=4, K=0.4
Du = 0.1, Da = 10-4
13.4, K0.1
G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)
Enhancement of temporal regularity:
correlation time vs. noise amplitude
vs. feedback gain
Large effect for small noise intensity
=7: increase
=5: decrease
Du = 0.1, Da = 10-4
G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)
Coherence resonance

Correlation time:
t cor    ( s ) ds
 – normalized
Gang, Ditzinger, Ning, Haken,0PRL 71, 807 (1993) autocorrelation function
Pikovsky, Kurths, PRL 78, 775 (1997)
Excitable System
a=1.1
=0.01
Simplified FitzHugh-Nagumo (FHN) system: excitable neuron
Example of coherence resonance: neuron
Time series of the membrane potential for
various noise intensity:
Simulation from
S.-G. Lee, A. Neiman, S. Kim,
PRE 57, 3292 (1998).
FitzHugh-Nagumo model with delay
Excitability
a=1:
excitability
threshold
3
  0.01; a  1.1;   4.12
x
x  x 
y
3
y  x  a  K( y   y)  D( t )
u activator
(membrane voltage)
v inhibitor
(recovery variable)
e time-scale ratio
Janson, Balanov, Schöll, PRL 93, 010601 (2004)
Coherence vs.  and K
D=0.09
D=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 1, D1 ,
2, D2
a= 1.05, 1=0.005, 2= 0.1, D2=0.09 : coherence resonance as function of D1
B. Hauschildt, N. Janson, A. Balanov, E. Schöll, PRE 74, 051906 (2006)
Stochastic synchronization
●
●
Frequency synchronization : mean interspike intervals (ISI)
Phase synchronization:
1:1 synchronization index
(Rosenblum et al 2001)
X
o
+
+ weakly synchronized
o moderately synchronized
x strongly synchronized
Local delayed feedback control:
enhance or suppress synchronization
System 1
●
Moderately synchronized system (o)
1:1 synchronization index
Delayed coupling, no self-feedback + noise
induces
antiphase
oscillations
Dahlem,
Hiller, Panchuk,
Schöll, IJBC in
print, 2008
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
Average phase synchronization time:
Schöll, Hiller,
Hövel, Dahlem,
2008
excitability
parameter
a=1.3,
oscillatory
regime,
C=K=0.5
Spreading depolarization wave
(cortical spreading depression SD)
Examples:
●
migraine aura (visual halluzinations)
●
stroke
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
activator diffusion
_
u activator (membrane voltage)
v inhibitor (recovery variable)
Du diffusion coefficient
e time-scale ratio of inhibitor and activator
variables
b excitability parameter
Dahlem, Schneider, Schöll, Chaos (2008)
Transient excitation: tissue at risk (TAR)
pulses die out after some distance
different values of b and e
Dahlem, Schneider, Schöll, J. Theor. Biol. 251, 202 (2008)
Boundary of propagation of traveling
excitation pulses (SD)
pulse
excitable:
traveling pulses
Propagation verlocity
non-excitable:
transient
FHN system with feedback
Non-local, time-delayed feedback:
Instantaneous long-range feedback:
Time-delayed local feedback:
(electrophysiological activity)
(neurovascular coupling)
Dahlem et al
Chaos (2008)
Non-local feedback: suppression of CSD
vu
uu
Tissue at risk
uv
vv
Non-local feedback:
shift of propagation boundary
pulse width Dx
K=+/-0.2
Time-delayed feedback: suppression of SD
uu
vu
Tissue at risk
uv
vv
Time-delayed feedback:
shift of propagation boundary
uu
vu
vv
K=+/-0.2
pulse width Dt
vu
Conclusions
u Delayed feedback control of excitable systems
u Control of coherence and spectral properties
u Stabilization of chaotic deterministic patterns
u 2 coupled neurons as network motif
u FitzHugh-Nagumo system: suppression or enhancement of
stochastic synchronization by local delayed feedback
u Modulation by varying delay time
u Delay-coupled neurons:
u delay-induced antiphase oscillations of tunable frequency
u delayed self-feedback: synchronization of oscillation modes
u Failure of feedback as mechanism of spreading
depression
u non-local or time-delayed feedback suppresses propagation of
excitation pulses for suitably chosen spatial connections or
time delays
Postdoc
Markus Dahlem
Students
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Roland Aust
Thomas Dahms
Valentin Flunkert
Birte Hauschildt
Gerald Hiller
Johanne Hizanidis
Philipp Hövel
Niels Majer
Felix Schneider
Collaborators
Andreas Amann
Alexander Balanov
Bernold Fiedler
Natalia Janson
Wolfram Just
Sylvia Schikora
Hans-Jürgen Wünsche
Published
October 2007