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 Published October 2007 Scope has considerably widened 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 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 timeĭelay 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 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
Net-Works 2008 Pamplona 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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