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 Net-Works 2008 Pamplona 10.6.2008
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TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS
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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
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
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
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)
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
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
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