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TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS

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.

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TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS

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  1. 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

  2. Outline iIntroduction: Time-delayed feedback control of nonlinear systems 8control of deterministic states 8control of noise-induced oscillations 8application: lasers, semiconductor nanostructures iNeural systems: control of coherence of neurons and synchronization of coupled neurons 8delay-coupled neurons8delayed self-feedback iControl of excitation pulses in spatio-temporal systems: migraine, stroke 8non-local instantaneous feedback 8time-delayed feedback

  3. Why is delay interesting in dynamics? iDelay increases the dimension of a differential equation to infinity: delay t generates infinitely many eigenmodes iSimple equation produces very complex behavior iDelay has been studied in classical control theory and mechanical engineering for a long time

  4. 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) ibiological systems: cell cycle time biological clocks 8neural networks: delayed coupling, delayed feedback

  5. 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 iApplication to spatio-temporal patterns: 8Partial differential equations

  6. Published October 2007 Scope has considerably widened

  7. Time-delayed feedback control of deterministic systems deterministic chaos t=T Stabilisation of unstable periodic orbits or unstable fixed points or space-time patterns iTime-delayed feedback (Pyragas 1992): Time-delay autosynchronisation (TDAS) t=T Extended time-delay autosynchronisation (ETDAS) (Socolar et al 1994) Many other schemes

  8. Time-delayed feedback control of deterministic systems stability is measured by Floquet exponent L: dx ~ exp(Lt) or Floquet multiplier m=exp(LT)

  9. Beyond Odd Number Limitation b complex (1 - gl)

  10. Example of all-optical time-delayed feedback control in semiconductor laser | Optical feedback: 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

  11. Stabilization of cw emission: Domain of control of unstable fixed point above Hopf bifurcation t=0.9T0 t=0.5T0 | Generic model: phase sensitive coupling l=0.2/T0 =0.2 w/2p Schikora, Hövel, Wünsche, Schöll, Henneberger , PRL 97, 213902 (2006)

  12. | Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006) Experimental realization

  13. DBRT Chemical reaction-diffusion systems Electrochemical systems Semiconductor nanostructures Hodgkin-Huxley neural models Examples: R • Global coupling due to Kirchhoff equation: I Control of spatio-temporal patterns: semiconductor nanostructure Without control: a(x,t): activator variable u(t): inhibitor variable f(a,u): bistable kinetic function D(a): transverse diffusion coefficient | Global coupling: Ratio of timescales: e Control parameters: e = RC, U0

  14. umin , umax 9.1 u Chaotic breathing pattern e = 9.1: above period doubling cascade Spatially inhomogeneous chaotic oscillations J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)

  15. umin , umax Stabilisation of unstable period-1 orbit tracking • Period doubling bifurcations generate a family of unstable periodic orbits (UPOs) • Period-1 orbit: Breathing oscillations Resonant tunneling diode a(x,t): electron concentration in quantum well u(t): voltage across diode

  16. Time-delayed feedback control of noise-induced oscillations deterministic chaos noise-induced oscillations t=T no deterministic orbits! Stabilisation of UPO t=T ? K. Pyragas, Phys. Lett. A 170, 421 (1992) N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)

  17. 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 t = 0.5TRO Flunkert and Schöll, PRE 76, 066202 (2007)

  18. Feedback control of noise-induced space-time patterns in the DBRT nanostructure t=13.4, K=0.1 t=4, K=0.4 Du = 0.1, Da = 10-4 G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)

  19. Enhancement of temporal regularity:correlation time vs. noise amplitude vs. feedback gain Large effect for small noise intensity t=7: increase t=5: decrease Du = 0.1, Da = 10-4 G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)

  20. Coherence resonance Excitable System a=1.1 e=0.01 Simplified FitzHugh-Nagumo (FHN) system: excitable neuron Correlation time: Y(t) – normalized autocorrelation function Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993) Pikovsky, Kurths, PRL 78, 775 (1997)

  21. Example of coherence resonance: neuron Time series of the membrane potentialfor various noise intensity: Simulation from S.-G. Lee, A. Neiman, S. Kim, PRE 57, 3292 (1998).

  22. FitzHugh-Nagumo model with delay Excitability a=1: excitability threshold u activator (membrane voltage) v inhibitor (recovery variable) e time-scale ratio Janson, Balanov, Schöll, PRL 93, 010601 (2004)

  23. D=0.09 Coherence vs. t and K 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)

  24. 2 non-identical stochastic oscillators: diffusive coupling frequencies tuned by e1, D1 , e2, D2 2 coupled FitzHugh-Nagumo systems:coupled neuron model as network motif a= 1.05, e1=0.005, e2= 0.1, D2=0.09 : coherence resonance as function of D1 B. Hauschildt, N. Janson, A. Balanov, E. Schöll, PRE 74, 051906 (2006)

  25. Frequency synchronization : mean interspike intervals (ISI) Phase synchronization: 1:1 synchronization index (Rosenblum et al 2001) Stochastic synchronization o X + + weakly synchronized o moderately synchronized x strongly synchronized

  26. Moderately synchronized system (o) Local delayed feedback control: enhance or suppress synchronization System 1 1:1 synchronization index

  27. Delayed coupling, no self-feedback + noise induces antiphase oscillations Dahlem, Hiller, Panchuk, Schöll, IJBC in print, 2008

  28. Sustained oscillations induced by delayed coupling excitability parameter a=1.3 a=1.05

  29. Regime of oscillations excitability parameter a=1.3

  30. Delayed coupling and delayed self-feedback Average phase synchronization time: excitability parameter a=1.3, oscillatory regime, C=K=0.5 Schöll, Hiller, Hövel, Dahlem, 2008

  31. migraine aura (visual halluzinations) stroke Spreading depolarization wave(cortical spreading depression SD) Examples:

  32. Migraine aura: neurological precursor(spatio-temporal pattern on visual cortex)

  33. Migraine aura: visual halluzinations

  34. Migraine aura: visual halluzinations

  35. Migraine aura: visual halluzinations

  36. Migraine aura: visual halluzinations

  37. Migraine aura: visual halluzinations

  38. Migraine aura: visual halluzinations

  39. Measured cortical spreading depression Visual cortex 3 mm/ min

  40. FitzHugh-Nagumo (FHN) system with activator diffusion _ u activator (membrane voltage) v inhibitor (recovery variable) Dudiffusion coefficient e time-scale ratio of inhibitor and activator variables b excitability parameter Dahlem, Schneider, Schöll, Chaos (2008)

  41. 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)

  42. Boundary of propagation of traveling excitation pulses (SD) pulse Propagation verlocity excitable: traveling pulses non-excitable: transient

  43. FHN system with feedback Non-local, time-delayed feedback: Instantaneous long-range feedback: (electrophysiological activity) (neurovascular coupling) Time-delayed local feedback: Dahlem et al Chaos (2008)

  44. Non-local feedback: suppression of CSD vu uu Tissue at risk vv uv

  45. Non-local feedback:shift of propagation boundary pulse width Dx K=+/-0.2

  46. Time-delayed feedback: suppression of SD uu vu Tissue at risk vv uv

  47. Time-delayed feedback:shift of propagation boundary uu vu pulse width Dt vu vv K=+/-0.2

  48. Conclusions uDelayed feedback control of excitable systems uControl of coherence and spectral properties uStabilization of chaotic deterministic patterns u2 coupled neurons as network motif uFitzHugh-Nagumo system: suppression or enhancement of stochastic synchronization by local delayed feedback uModulation by varying delay time uDelay-coupled neurons: u delay-induced antiphase oscillations of tunable frequency u delayed self-feedback: synchronization of oscillation modes uFailure 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

  49. Roland Aust Thomas Dahms Valentin Flunkert Birte Hauschildt Gerald Hiller Johanne Hizanidis Philipp Hövel Niels Majer Felix Schneider Postdoc Students Markus Dahlem Collaborators Andreas Amann Alexander Balanov Bernold Fiedler Natalia Janson Wolfram Just Sylvia Schikora Hans-Jürgen Wünsche

  50. Published October 2007

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