A brief history of sync Folk theorems, myths, & conjectures Christiaan Huygens (1629 – 1695) in complex oscillator networks physicist & mathematician engineer & horologist NetSci 2015 Satellite Symposium observed “ an odd kind of sympathy ” Florian D¨ orfler [Letter to Royal Society of London, 1665] Recent reviews, experiments, & analysis [M. Bennet et al. ’02, M. Kapitaniak et al. ’12] 2 / 27 A field was born Coupled phase oscillators sync in mathematical biology [A. Winfree ’80, S.H. Strogatz ’03, . . . ] ∃ various models of oscillators & interactions sync in physics and chemistry [Y. Kuramoto ’83, M. M´ ezard et al. ’87. . . ] Today: coupled phase oscillator model sync in neural networks [F.C. Hoppensteadt and E.M. Izhikevich ’00, . . . ] [A. Winfree ’67, Y. Kuramoto ’75] sync in complex networks [C.W. Wu ’07, S. Bocaletti ’08, . . . ] � n ˙ θ i = ω i − j =1 a ij sin( θ i − θ j ) . . . and numerous technological applications (reviewed later) ω 2 ω 3 Physics Reports 469 (2008) 93–153 ◮ n oscillators with phase θ i ∈ S 1 Contents lists available at ScienceDirect Physics Reports a 23 journal homepage: www.elsevier.com/locate/physrep ◮ non-identical natural frequencies ω i ∈ R 1 a 12 Synchronization in complex networks a 13 ◮ elastic coupling with strength a ij = a ji Alex Arenas a,b , Albert Díaz-Guilera c,b , Jurgen Kurths d,e , Yamir Moreno b,f, ∗ , Changsong Zhou g a ◮ undirected & connected graph G = ( V , E , A ) ω 1 Note: can be derived as canonical coupled limit-cycle oscillator model 3 / 27 4 / 27
My application of interest: sync in AC power networks My application of interest: sync in AC power networks sync is crucial for AC power grids – a coupled oscillator analogy sync is crucial for AC power grids – a coupled oscillator analogy sync is a trade-off sync is a trade-off weak coupling & heterogeneous strong coupling & homogeneous weak coupling & heterogeneous Blackout India July 30/31 2012 5 / 27 5 / 27 Other technological applications of phase oscillators Phenomenology and challenges in synchronization many fundamental questions are still open particle filtering to estimate limit cycles [A. Tilton & P. Mehta et al. ’12] Transition to synchronization is a trade-off: coupling vs. heterogeneity clock synchronization over networks ∆Φ 1 ( t ) PD ε ( s ) [Y. Hong & A. Scaglione ’05, O. Simeone et s 1 ( t ) VCO ∆Φ 3 ( t ) PD 1 al. ’08, Y. Wang & F. Doyle et al. ’12] ( s ) ε T 1 s 3 ( t ) VCO ∆Φ 2 ( t ) PD ε ( s ) 1 central pattern generators and T 3 s 2 ( t ) VCO robotic locomotion [J. Nakanishi et al. 1 T 2 ’04, S. Aoi et al. ’05, L. Righetti et al. ’06] decentralized maximum likelihood quantify “coupling” vs. “heterogeneity” Some central questions: estimation [S. Barbarossa et al. ’07] multiple sync’d states & their sync basin (still after 45 years of work) carrier sync without phase-locked interplay of network & dynamics loops [M. Rahman et al. ’11] In more technical terms: existence, uniqueness, & stability of equilibria and robotic vehicle coordination θ ( x, y ) their basin of attraction . . . as a function of network topology & parameters [R. Sepulchre et al. ’07, D. Klein et al. ’09] θ 6 / 27 7 / 27
Outline Main references today Automatica 50 (2014) 1539–1564 Contents lists available at ScienceDirect Introduction Automatica journal homepage: www.elsevier.com/locate/automatica Synchronization Threshold Survey paper Synchronization in complex networks of phase oscillators: A survey I Equilibrium Landscape Florian Dörfler a,1 , Francesco Bullo b a Automatic Control Laboratory, ETH Zürich, Switzerland b Department of Mechanical Engineering, University of California Santa Barbara, USA Almost Global Synchronization CHAOS 25 , 053103 (2015) Conclusions Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis Dhagash Mehta, 1,a) Noah S. Daleo, 2,b) Florian D € orfler, 3,c) and Jonathan D. Hauenstein 1,d) I try to shed light on some fundamental yet poorly understood questions. 1 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556, USA 2 Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695, USA 3 Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH) Z € urich, 8092 Z € urich, Switzerland 8 / 27 Models & sync notion finite dimensional & heterogeneous uniform all-to-all Kuramoto model general coupled oscillator model � n K � n ˙ ˙ θ i = ω i − j =1 a ij sin( θ i − θ j ) θ i = ω i − n sin( θ i − θ j ) j =1 the synchronization threshold where K > 0 is the coupling where a ij = a ji ≥ 0 induces a strength among the oscillators connected and undirected graph or existence, uniqueness, & Frequency synchronization: ˙ θ i = ω sync ∈ R for all i ∈ { 1 , . . . , n } local stability of equilibria Lemma: if there is a frequency-sync’d solution, then ω sync = � n i =1 ω i / n ⇒ frequency-synchronized solutions are equilibria in rotating coordinates 9 / 27
Synchronization threshold for the complete graph Synchronization threshold for the complete graph – cont’d 1 explicit & tight lower/upper bounds [Chopra & Spong ’09, FD & Bullo ’11] K � n ˙ θ i = ω i − n sin( θ i − θ j ) synchronization if K > K crit ( ω ) j =1 1 2 max i , j | ω i − ω j | ≤ K crit ≤ max i , j | ω i − ω j | 1 ⇒ necessary & tight lower bound [Chopra & Spong ’09] K crit ≥ max 2 | ω i − ω j | 2 exact & implicit [Aeyels & Rogge ’04, Mirollo & Strogatz ’05, Verwoerd & Mason ’08] i , j 1 − ( ω i / u ∗ ) 2 where u ∗ ∈ [ � ω � ∞ , 2 � ω � ∞ ] is the unique nu ∗ √ K crit = � n i =1 K crit ≤ max i , j | ω i − ω j | ⇒ sufficient & tight upper bound [FD & Bullo ’11] 1 − ( ω i / u ) 2 = � n 1 − ( ω i / u ) 2 . solution to the equation 2 � n � � i =1 1 / i =1 g trip, n ( ω ) g bip ( ω ) 2 Kuramoto’s n − 2 1.8 upper explicit p continuum π / 2 comparison of bounds n π / 2 π / 2 1.6 limit bound exact & implicit for uniform distribution K crit (1 − p ) 1.4 4 / π 1 1 1.2 g unif ( ω ) ∈ [ − 1 , +1] n n 1 lower explicit + ω 0 − ω 0 0 ω ω m in ω max ω 0 0.8 0.6 tight lower bound tight upper bound 1 2 10 / 27 11 / 27 10 10 n Primer on algebraic graph theory Laplacian matrix L = “degree matrix” − “adjacency matrix” . . . ... ... . . . . . . L = L T = � n ≥ 0 there’s nothing more to say − a i 1 · · · j =1 a ij · · · − a in . . . ... ... . . . . . . for the complete uniform graph Notions of connectivity . . . so let’s move on spectral: 2nd smallest eigenvalue of L is “algebraic connectivity” λ 2 ( L ) topological: degree � n j =1 a ij or degree distribution Notions of heterogeneity { i , j }∈E | ω i − ω j | 2 � 1 / 2 � � � ω � E , ∞ = max { i , j }∈E | ω i − ω j | , � ω � E , 2 = 12 / 27
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